pdfs.semanticscholar.orgpdfs.semanticscholar.org/6756/b1637645087fa642c7bfa4d5f... · 2019-01-02 · SIAM J. SCI. COMPUT. c 2014 Society for Industrial and Applied Mathematics Vol.

SIAM J. SCI. COMPUT. c© 2014 Society for Industrial and Applied MathematicsVol. 36, No. 4, pp. A1359–A1383

NESTED NEWTON STRATEGIES FOR ENERGY-CORRECTEDFINITE ELEMENT METHODS∗

U. RUDE† , C. WALUGA‡ , AND B. WOHLMUTH‡

Abstract. Energy-corrected finite element methods provide an attractive technique for dealingwith elliptic problems in domains with re-entrant corners. Optimal convergence rates in weighted L2-norms can be fully recovered by a local modification of the stiffness matrix at the re-entrant corner,and no pollution effect occurs. Although the existence of optimal correction factors is established,it remains open how to determine these factors in practice. First, we show that asymptotically aunique correction parameter exists and that it can be formally obtained as the limit of level dependentcorrection parameters which are defined as roots of an energy defect function. Second, we proposethree nested Newton-type algorithms using only one Newton step per refinement level and show localor even global convergence to this asymptotic correction parameter.

Key words. corner singularities, energy-corrected finite element methods, optimal convergencerates, pollution effect, re-entrant corners

AMS subject classifications. 65N30, 65N12, 65N15

DOI. 10.1137/130935392

1. Introduction. Elliptic partial differential equations with singular solutioncomponents are of special interest in many applications, e.g., in fracture mechanics orin heterogeneous porous media flow. In the presence of singular solution components,standard finite element methods on uniform meshes show poor results compared to thebest approximation in the L2-norm. This effect is commonly referred to as pollutionand is related to the fact that the solution of the dual problem has, in general, noH2-regularity.

In this work, we consider the Laplace problem in a two-dimensional domain Ωwith a re-entrant corner as a prototype for a systematic approach to pollution effects.We refer the reader to [21] for a numerical study of more general settings, includingsecond order finite elements, eigenvalue problems, and heterogeneous coefficients. Itis well known (see, e.g., [17, 18, 22, 26]) that in the presence of corners with interiorangle π < θ < 2π the solution will, in general, have singular components of typerkπ/θ sin(kπφ/θ), k ∈ N, even when the data are smooth. Here, r denotes the distanceto the singular corner and φ the angle. Thus, the singular components of the solutionare not smooth, i.e., s := rπ/θ sin(πφ/θ) �∈ Hα(Ω), α ≥ 1 + π/θ. As a consequence,only reduced convergence rates are obtained by standard finite element methods on a

sequence of quasi-uniform meshes; i.e., the estimates ‖∇(u − Rlu)‖0 = O(hπ/θl ) and

‖u − Rlu‖0 = O(h2π/θl ) for the Ritz projector Rl are, in general, sharp with respect

to powers of the mesh-size hl. Although the convergence order in the H1-norm isthe same as the order of the best approximation, this does not hold for the L2-norm,where a gap of 1 − π/θ in the convergence order can be observed [6, 7, 30]. For a

∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section September3, 2013; accepted for publication (in revised form) April 4, 2014; published electronically July 1,2014.

http://www.siam.org/journals/sisc/36-4/93539.html†Department of Computer Science 10, University Erlangen-Nuremberg, Cauerstraße 6, D–91058

Erlangen, Germany ([email protected]).‡Institute for Numerical Mathematics, Technische Universitat Munchen, Boltzmannstrasse 3, D–

85748 Garching b. Munchen, Germany ([email protected], [email protected]).

A1359

http://www.siam.org/journals/sisc/36-4/93539.html

mailto:[email protected]



A1360 U. RUDE, C. WALUGA, AND B. WOHLMUTH

weighted L2-norm with weight r1−π/θ, the gap is even bigger. The nodal interpolantthen shows O(h2) convergence while such a weight does not recover any additionalconvergence rate for the Ritz projector, i.e., typically a gap of 2(1−π/θ) can be seen.

Standard techniques to improve the convergence are to deal with weighted Sobolevspace norms in combination with graded meshes [1, 2, 10, 12, 13, 15] and/or enrich-ment of the finite element space [3, 6, 9, 11, 14, 20, 25, 30]. Alternative techniques arefirst order systems least squares methods that provide the flexibility to select appro-priate weights in the norms; see, e.g., [4, 5, 8, 16, 23]. Most strategies require morethan a local modification and aim to improve the finite element approximation at thesingularity. However, the quantity of interest is often not a global standard norm buta functional or weighted norm which excludes or relaxes the influence of the neigh-borhood of the re-entrant corner. Examples are the stress intensity factor that can beevaluated as a line integral at a fixed distance from the re-entrant corner, the eigen-values, and the flux at some given interface not including the re-entrant corner. Toobtain improved error reduction rates for such quantities, an accurate representationof the solution at the re-entrant corner is not required.

This observation is our motivation to focus on energy correction schemes [21,28, 29] that do not enrich the finite element spaces associated with a sequence ofuniformly refined meshes. The basic idea was originally introduced for finite differenceschemes in [33]. In the recent contribution [19], a mathematically rigorous analysisfor finite element methods is presented, and it was shown that a careful modificationof the original Galerkin method can drastically improve the convergence. Namely, forconforming low-order finite element spaces, if π < θ < 3π/2, then O(h2

l ) convergenceon a sequence of uniformly refined meshes can be observed in a suitable weightedL2-norm, assuming that a sufficiently accurate correction parameter is known. Ifadditionally the initial mesh restricted to elements touching the re-entrant corner issymmetric, the same argument also holds for 3π/2 ≤ θ < 2π.

The rest of this paper is organized as follows: First, in section 2, we sketchthe energy corrected finite element method and motivate the need for an efficientalgorithm to determine its parameter. In section 3, we introduce a level-dependentnonlinear energy defect function and recall that its unique root defines a suitablelevel-dependent parameter. Based on the properties of the energy defect, we establishglobal convergence of a Newton algorithm and give reliable stopping criteria. Section4 is devoted to the proof that the roots of these functions converge, and we provideconvergence rates. Additionally, we show that the limit value defines the uniquelevel-independent parameter in energy-corrected finite element methods. We discuss,in section 5, several nested Newton-type strategies to approximate the roots and showthat the asymptotic correction parameter can be approximated with almost no extracost. Numerical results that illustrate the convergence of the different strategies canbe found in section 6. Here we also investigate the dependency of the asymptoticcorrection factor on the angle of the re-entrant corner and the number of attachedelements. Additionally, we apply the approach to a domain with several re-entrantcorners and more general boundary conditions.

2. Energy-corrected finite element method. In this section, we sketch theidea of energy-corrected finite element methods. For simplicity we restrict ourselvesto a simple model problem and to a bounded polygonal domain Ω ⊂ R

2 with onere-entrant corner. We consider the numerical solution of the Poisson problem

−Δu = f in Ω, u = 0 on ∂Ω.(2.1)

NESTED NEWTON FOR ENERGY-CORRECTED FINITE ELEMENTS A1361

The standard bilinear form associated with (2.1) is given by a(v, w) :=∫Ω∇v·∇w dx,

v, w ∈ H1(Ω), and (·, ·) denotes the usual L2-scalar product.

2.1. Definition of the energy-corrected finite element method. To definethe energy correction, we introduce

al(v, w) :=

∫ωl

∇v ·∇w dx,

where ωl ⊂ Bk0hlis a union of elements in Tl. The sequence Tl forms a nested

hierarchy of uniformly refined simplicial meshes with mesh-size hl, and Bk0hlis a ball

with radius k0hl and center at the re-entrant corner. Asymptotically, it is essentialthat k0 be fixed and bounded independently of the level l, since this guarantees thatthe number of elements in ωl is bounded independently of l; see subsection 3.3 for atheoretical discussion. We note that Algorithm 1 of subsection 3.5 gives us for all ournumerical examples k0 = 1. For given γ ∈ [0, 0.5], we define the parameter-dependentbilinear form by

(2.2) aec(v, w) := a(v, w)− γal(v, w)

and note that it depends on the mesh-dependent subdomain ωl and also on the scalarparameter γ. Obviously, we have 0.5 a(v, v) ≤ aec(v, v) ≤ a(v, v) for v ∈ H1(Ω). Theassumption γ ∈ [0, 0.5] guarantees that aec(·, ·) is uniformly coercive on H1

0 (Ω).A modified finite element formulation of (2.1) then reads as follows: Find ul(γ) ∈

Vl ∩H10 (Ω) such that

(2.3) aec(ul(γ), v) = (f, v), v ∈ Vl ∩H10 (Ω).

Here, Vl stands for the standard conforming piecewise linear finite element space asso-ciated with Tl. The modification (2.3) does not change the structure of the standardfinite element stiffness matrix and changes only a small number of its coefficients.Hence, it is cheap and easy to implement into existing codes, provided that γl andωl are given. Moreover, fast high performance solvers may profit from using datastructures for uniformly refined grids that avoid the logistic overhead of unstructuredand adaptive mesh techniques.

Remark 2.1. We note that for γ = 0 the standard finite element solution isrecovered. Recalling that the Poisson equation models the normal displacement ofa homogeneous membrane, the effect of the modification with γ ∈ (0, 0.5] can beregarded as a softening of the material in ωl.

We emphasize that the quality of ul(γ) is determined by the choice of γ and ωl.The choice of ωl is motivated by the fact that the modification should change the orig-inal stiffness matrix as little as possible and should not deteriorate the convergenceorder. In [19] it has been shown that such ωl and a level-dependent parameter γ existsuch that no pollution occurs; i.e., there is no gap in the convergence order betweenthe interpolation and energy-corrected finite element approximation. Moreover, sec-ond order convergence in a suitably weighted L2-norm can be recovered. A suitablecorrection parameter γ can be defined by γ := γl, where γl is the root of a nonlinearenergy defect function that will be introduced in section 3. In all our numerical ex-amples of section 6 and also those discussed in [19, 21], it has been sufficient to choosethe correction domain ωl as the union of elements adjacent to the re-entrant corner.We now call the modified finite element approach (2.3) the energy-corrected finiteelement method if ωl and the possibly level-dependent parameter γ are selected suchthat optimal order convergence rates can be observed on uniformly refined meshes.


2.2. Numerical example. We start with an illustrating example that showsthe performance of an energy-corrected finite element method. Moreover from thisexample it will become apparent how important the proper selection of the parameterγ is. To demonstrate the accuracy of the energy-correction method and to motivatethe algorithms proposed in this paper, we consider here a triangulation of a polygonaldomain Ω with a single re-entrant corner at xc = (0, 0) and the interior angle θ = 3π/2(L-shape). We set nonhomogeneous Dirichlet boundary conditions given in polarcoordinates as u = s := r2/3 sin(2φ/3) on ∂Ω and zero forcing f = 0 such that theexact solution u = s ∈ H1+π/θ−ε(Ω) for any ε > 0. Besides the standard norms, weemploy the weighted L2-norm (L2

ρ) and the weighted H1-seminorm (H1ρ) defined by

(2.4) ‖u− uh‖0,ρ := ‖ρ(u− uh)‖0, |u− uh|1,ρ := ‖ρ∇(u− uh)‖0,

with the radial weighting function defined as ρ := r1−π/θ. This weight is illustratedin Figure 1 along with the initial triangulation and the solution.

Fig. 1. Initial mesh, weighting function, and solution for the L-shape example.

We next conduct a convergence study for different values of the correction pa-rameter γ on a series of uniformly refined meshes. All finite element formulationsshow almost the same quantitative results in the H1-norms, but there is a significantdifference in the performance for the L2-norms. Table 1 presents the errors for thestandard finite element method when no correction is used (γ = 0). As is well known,this shows the pollution effect, i.e., it results in suboptimal convergence rates, in bothweighted and standard L2-norms.

Table 1

Standard and weighted L2 errors for the L-shape domain with γ = 0.0.

l L2 error Rate L2ρ error Rate H1 error Rate H1

ρ error Rate

0 3.704e-2 - 2.362e-2 - 2.506e-1 - 1.625e-1 -

1 1.434e-2 1.37 8.248e-3 1.52 1.622e-1 0.63 8.677e-2 0.91

2 5.545e-3 1.37 3.002e-3 1.46 1.039e-1 0.64 4.595e-2 0.92

3 2.150e-3 1.37 1.129e-3 1.41 6.620e-2 0.65 2.414e-2 0.93

4 8.370e-4 1.36 4.336e-4 1.38 4.199e-2 0.66 1.259e-2 0.94

5 3.273e-4 1.35 1.688e-4 1.36 2.657e-2 0.66 6.535e-3 0.95

Next, we guess a correction parameter γ = 0.1 for which we observe a significantimprovement of the solution accuracy in the L2-norms as shown in Table 2. However,here the asymptotic behavior remains suboptimal. This indicates that the guessedcorrection parameter may be considered good enough for the first two mesh levelsl = 1, 2 but that it is not sufficiently accurate for higher levels. Extending the results


Table 2

Standard and weighted L2 errors for the L-shape domain with γ = 0.1.


ρ error Rate

0 3.016e-2 - 1.784e-2 - 2.541e-1 - 1.645e-1 -

1 9.722e-3 1.63 4.664e-3 1.94 1.652e-1 0.62 8.801e-2 0.90

2 3.140e-3 1.63 1.257e-3 1.89 1.060e-1 0.64 4.653e-2 0.92

3 1.019e-3 1.62 3.564e-4 1.82 6.754e-2 0.65 2.439e-2 0.93

4 3.339e-4 1.61 1.086e-4 1.71 4.284e-2 0.66 1.270e-2 0.94

5 1.108e-4 1.59 3.597e-5 1.59 2.711e-2 0.66 6.577e-3 0.95

of [19], we will show in section 3.4 that on each mesh level there exists a suitableinterval of energy correction parameters. Asymptotically there additionally exists aunique parameter γ∞ which works for all levels. We will design efficient algorithmsto approximate this limit value accurately enough, which in this example is given byγ∞ = 0.11917674 . . . . Using this asymptotically correct parameter for computing theresults in Table 3 yields the optimal convergence rates predicted by the theory forboth weighted and standard L2-norms.

Table 3

Standard and weighted L2 errors for the L-shape domain with γ = γ∞.


ρ error Rate

0 2.911e-2 - 1.700e-2 - 2.557e-1 - 1.659e-1 -

1 9.147e-3 1.67 4.229e-3 2.01 1.666e-1 0.62 8.887e-2 0.90

2 2.882e-3 1.67 1.057e-3 2.00 1.069e-1 0.64 4.696e-2 0.92

3 9.084e-4 1.67 2.646e-4 2.00 6.815e-2 0.65 2.459e-2 0.93

4 2.862e-4 1.67 6.619e-5 2.00 4.323e-2 0.66 1.279e-2 0.94

5 9.021e-5 1.67 1.656e-5 2.00 2.735e-2 0.66 6.620e-3 0.95

However, the question of how to compute the parameter γ∞ or how to find asuitable level-dependent parameter interval remains open. Unfortunately, there isso far no analytical formula known to determine these quantities, which depend onthe angle and on the initial mesh at the re-entrant corner but not on the globalmesh and not on the solution. The main contribution of this paper is to develop andanalyze Newton-type algorithms for the approximation of γ∞. Moreover we show thatthe computed approximations on each level define an energy-corrected finite elementmethod. The three algorithms proposed in section 5 each require one step of a Newtoniteration per refinement level of the mesh to determine such a correction parameter.They differ in whether the analytic exact energy of the dominating singular functionmust be known and in how many finite element systems must be solved per step.

3. The energy defect function. For the ease of presentation, we assume againthat Ω ⊂ R

2 has one re-entrant corner xc = (0, 0), that a part of the positive x-axis starting at the origin is in ∂Ω, and that all x ∈ Ω can be represented in polarcoordinates as x = (r cosφ, r sinφ) with φ ∈ [0, θ]. Let s := rπ/θ sin(πφ/θ); then s isa solution of the Dirichlet boundary value problem: Find u such that

−Δu = 0 in Ω, u = s on ∂Ω.(3.1)

Moreover a(s, v) = 0 for all v ∈ H10 (Ω).

In terms of the bilinear form aec(·, ·) given in (2.2), we define a finite elementapproximation sl(γ) ∈ Vl such that the inhomogeneous Dirichlet boundary conditions


sl(γ)(pl) = s(pl) are satisfied for all vertices of Tl being on ∂Ω and

(3.2) aec(sl(γ), vl) = 0, vl ∈ Vl ∩H10 (Ω).

We recall that for γ ∈ [0, 0.5], the bilinear form aec(·, ·) is uniformly elliptic, and thusa unique solution exists. Moreover, we have

(3.3) ‖∇(s− sl(γ))‖0;Ω\B ≤ Chl, ch2π/θl ≥ ‖∇(s− sl(γ))‖20 ≥ 2C0h

2π/θl ,

where B denotes a ball with a fixed and positive diameter and center at the re-entrant corner. The first bound follows from the regularity of s on Ω \B, the globalL2-estimate ‖s− sl(γ)‖0 = O(hl), and Wahlbin-type arguments for the finite elementerror on subdomains [32]. The second equivalence follows from the best approximationproperties and the properties of the modified bilinear form aec(·, ·). We refer the readerto [19] for details.

Now we define on each level l the energy defect function gl(γ), for γ ∈ [0, 0.5], as

(3.4) gl(γ) := a(s, s)− aec(sl(γ), sl(γ)).

The main difference compared to [19] is that we define the energy defect functionin terms of the singular function s and not with respect to a cut-off of s havinghomogeneous Dirichlet boundary values on ∂Ω. Observing that sl(γ) restricted to ∂Ωdoes not depend on γ, we get that s′l(γ) ∈ Vl ∩ H1

0 (Ω), where the prime ′ stands forthe derivative with respect to γ. A straightforward computation then shows that

g′l(γ) = −2aec(sl(γ), s′l(γ)) + al(sl(γ), sl(γ)) = al(sl(γ), sl(γ)) ≥ 0,(3.5)

and thus gl(·) defined by (3.4) is, in contrast to the definition given in [19], a mono-tonically increasing function on [0, 0.5]. Moreover, it is easy to see that s′l(γ) satisfiesthe variational problem

(3.6) aec(s′l(γ), vl) = al(sl(γ), vl), vl ∈ Vl ∩H1

0 (Ω).

3.1. Algebraic representation of the energy defect function. For sim-plicity of notation, we suppress in the algebraic representation the level index l andindicate the γ dependence by an upper index. We decompose the nontrivial degreesof freedom of the algebraic representation of sl(γ) into two blocks. Then sl(γ) ∈ Vl

can be identified with the vector sγ having the two block components sγI and sγR. Thevalues of sγI are the nodal values of sl(γ) at the inner vertices, and the values of sγRare the nodal values of sl(γ) at a subset of boundary vertices. More precisely, let θpbe the angle of the polar coordinates of the boundary vertex p; then we include p inthe subset of boundary nodes if and only if θp ∈ (0, θ); see Figure 2.

We assemble the matrices with respect to the reduced space Vl := {vl ∈ Vl;vl(x) = 0, x ∈ ∂ΩR} with ∂ΩR := {x ∈ ∂Ω, x = (rx, 0) or x = (rx cos θ, rx sin θ)}, andnl := dim(Vl). By A we denote the standard stiffness matrix on level l associated withNeumann boundary conditions, by AD the stiffness matrix associated with Dirichletconditions, and by B the matrix associated with al(·, ·). All three stiffness matricesA,AD, B ∈ R

nl×nl have a 2×2 block structure associated with the degrees of freedomin the two blocks I and R. In this algebraic notation, the coefficient vector sγ of themodified finite element solution sl(γ) satisfies

(3.7) (AD − γB)sγ =:

((AII AIR

0 Id

)− γ

(B 00 0

))(sγIsγR

)=

(0sR,

),


Fig. 2. The entries in the block component I are marked with filled circles and the entries inthe block component R are marked with empty squares (N = 4 for k = 1).

where sR stands for the vector obtained by nodal interpolation of the singular functions on part of the boundary. Having (3.7), gl(γ) defined by (3.4) and g′l(γ) defined in(3.5) can be expressed equivalently by

gl(γ) = a(s, s)− s�R(ARR −ARI(AII − γB)−1AIR)sR,(3.8a)

g′l(γ) = s�RARI(AII − γB)−1B(AII − γB)−1AIRsR,(3.8b)

with ARI := A�IR. We note that the matrix B is symmetric and positive semidefinite

and of low rank N , where N is the number of interior vertices contained in ωl; seealso Figure 2.

The matrix AII is positive definite, and moreover a straightforward computationshows that

(3.9) x�Bx ≤ qx�AIIx,

with q = 1 for k0 > 1 and q < 1 for k0 = 1.

3.2. Relation to singular enrichment. As an alternative to the previousrank-N modification of the stiffness matrix, one could consider a rank-1 modifica-tion having the same local effect. This can be accomplished by first enriching Vl withone basis function φE compactly supported in Ω. More precisely, we require thatthe support of φE be bounded away from the outer boundary, suppφE ∩ T = ∅ if∂T ∩ (∂Ω \ ∂ΩR) �= ∅. Second, we consider the original variational problem on theenriched space and apply static condensation to obtain

(3.10)

((AII AIR

0 Id

)− γ

(S 00 0

))(sγIsγR

)=

(0sR

),

where the submatrix S has the form S := AIEA−1EEAEI and γ = 1. Here AEE :=

a(φE, φE) is a scalar. AIE = A�EI, and AEI is a row vector whose entries are obtained

by evaluating a(·, φE) with the basis functions from the I-block. Now we set γ ∈ [0, 1]as parameter. For γ = 0 we obtain the standard formulation and for γ = 1 theenriched form as described above. For γ ∈ (0, 1) we get by construction a rank-1modification acting as a local softening of the material. Moreover if φE is supportedin ωl, then S has the same sparsity pattern as B in (3.7), i.e., it has at most N2 non-zero entries. If φE is a cut-off of the singular function being supported in a subdomainindependently of h and γ = 1, we obtain the condensed form of a singular enrichment.In this case, however, the number of nonzero entries in S grows as the mesh-size tendsto zero.

Lemma 3.1. If the modification in its algebraic form is given by (3.10) with S =zz�, then the energy defect function has a unique root if β := z�y−αs�RARIyy

�AIRsR �=


0. Here α−1 := a(s, s) − a(sl(0), sl(0)) and y := A−1II z. Moreover the unique root is

then analytically given by γl =1β .

Proof. The proof follows by an application of the Sherman–Morrison formula in(3.8a) and the fact that γl is characterized by gl(γl) = 0. We recall that the Sherman–Morrison formula as a special case of the Sherman–Morrison–Woodbury formula readsas

(M + wv�)−1 = M−1 − M−1wv�M−1

1 + v�M−1w

for quite general matrices M and vectors w, v. Setting M := AII and w := −γz andv := z, we get that γl is defined by

a(s, s)− s�R

(ARR −ARI

(A−1

II −A−1

II (−γlz)z�A−1II

1 + z�A−1II (−γlz)

)AIR

)sR = 0.

Observing that a(sl(0), sl(0)) = s�R(ARR −ARIA−1II AIR)sR, a straightforward compu-

tation gives γl = β−1.

Based on Lemma 3.1 the computation of γl would cost the solution of two un-modified discrete boundary value problems. However, it does not provide the well-posedness and makes no statement about the convergence of such a γl. Thus inthe rest of this paper, we do not discuss this option any further but focus on themodification defined by (2.2) and provide efficient algorithms to approximate suitablecorrection parameters.

Remark 3.2. Equation (3.8a) shows that the nonlinearity in gl(·) stems from theterm (AII − γB)−1. Recalling that B is a low rank matrix, one can use a low rank

representation of (Id − γA−1/2II BA

−1/2II )−1 using the Sherman–Morrison–Woodbury

formula to rewrite gl(·) as a rational function in γ, i.e., gl(γ) = Pl(γ)/Ql(γ), wherePl and Ql are polynomials of degree at most N . The coefficients of these polynomialscan be computed numerically by solving N times a discrete boundary value problem.

In [19] it has been shown that optimal convergence order can be observed if theparameter γ on each level l is selected such that |gl(γ)| ≤ Ch2

l with C fixed andmoderate. In the rest of this section, we first provide existence results and thenpropose a Newton algorithm including a reliable stopping criteria for the selection ofωl and γ on each level.

3.3. Existence of ωl. The properties of gl(·) depend on the choice of ωl. Todetermine a suitable ωl, we follow along the lines of [19, 29, 28, 33]. Setting ω1

l :={T ∈ Tl, xc ∈ ∂T } and then recursively enlarging the neighborhood of xc by

ωk+1l := {T ∈ Tl, ∂T ∩ ωk

l �= ∅},

we define

γkl :=

|a(Ils, Ils)− a(s, s)|∫ωk

l∇Ils · ∇Ils dx

,

where Il is the standard nodal interpolation operator. Then it is obvious that γk+1l <

γkl . Considering the numerator in more detail, we obtain by integration by parts and


from the regularity of s the upper bound

|a(Ils, Ils)− a(s, s)| ≤ a(Ils− s, Ils− s) + 2|a(Ils− s, s)|

≤ ‖∇(s− Ils)‖20 + 2

∫∂Ω

∣∣∣∣ ∂s∂n (s− Ils)

∣∣∣∣ dτ≤ C(h

2π/θl + h2

l ) ≤ C21h

2π/θl ,

where in the last step we used that s = Ils = 0 on ∂ΩR ⊂ ∂Ω. The denominator canbe bounded from below in terms of the triangle inequality by

‖∇Ils‖0;ωkl≥ ‖∇s‖0;ωk

l− ‖∇(s− Ils)‖0;ωk

l

≥ ‖∇s‖0;ωkl− ‖∇(s− Ils)‖0 ≥

(C2k

π/θ − C1

)hπ/θl .

From now on we fix ωl := ωk0

l with k0 ∈ N large enough such that (C2kπ/θ0 − C1)

2 >

2C21 . This choice guarantees γk0

l < 0.5.

3.4. Existence of an interval for γ. We proceed in two steps. In Lemma 3.3,we show that gl(·) has a unique root in (0, 0.5). In Lemma 3.4, we establish lowerand upper bounds for g′l(·). These two preliminary results allow us to specify a closedinterval Jl such that for γ ∈ Jl, we obtain an energy-corrected finite element method.

Lemma 3.3. There exists a coarse level l0 ∈ N such that for l ≥ l0 there is aunique γl ∈ (0, 0.5) with gl(γl) = 0.

Proof. The proof is similar to the proof of [19, Lemma 5.2]; see also [28, Lemma3]. In a first step, we show existence and in a second step uniqueness. Since gl(·) iscontinuous, we start with the evaluation of gl(·) at γ = 0 and at γ = 0.5 and deducegl(0)gl(0.5) ≤ 0. For γ = 0, we find in terms of (3.3) and the fact that Δs = 0

gl(0) = a(s, s)− a(sl(0), sl(0)) = −a(s− sl(0), s− sl(0)) + 2a(s, s− sl(0))

≤ −2C0h2π/θl + 2

∫∂Ω

∂s

∂n(s− Ils)dτ ≤ −2C0h

2π/θl + C3h

2l

≤ h2π/θl (C3h

2(1−π/θ)l − 2C0).

Thus for l0 large enough, we have C3h2(1−π/θ)l0

≤ C0 and

(3.11) gl(0) ≤ −C0h2π/θl , l ≥ l0.

Noting that sl(γ) − Ils ∈ Vl ∩H10 (Ω) and thus aec(sl(γ), sl(γ) − Ils) = 0, we get for

γ = 0.5

gl(0.5) = a(s, s)− a(Ils, Ils) + 0.5 al(Ils, Ils) + ‖Ils− sl(0.5)‖2hl;0.5

≥ (0.5− γk0

l )al(Ils, Ils) + ‖Ils− sl(0.5)‖2hl;0.5> 0,

where ‖vl‖2hl;γ:= a(vl, vl) − γal(vl, vl). Due to the continuity of gl(·), there exists a

root γl ∈ (0, 0.5) for l large enough.To guarantee uniqueness, it is sufficient to sharpen the monotonicity of (3.5) and

show g′l(γl) �= 0. Assuming that al(sl(γl), sl(γl)) = 0, then sl(γl) restricted to ωl isequal to zero, and thus sl(γl) = sl(0). Now gl(γl) = 0, and gl(γl) = gl(0) yields acontradiction to gl(0) < 0, and thus gl(·) has one unique root.

Lemma 3.4. There exist two constants 0 < α ≤ β < ∞ and a level l0 such thatfor l ≥ l0 and γ ∈ [0, 0.5]


αh2π/θl ≤ g′l(γ) ≤ βh

2π/θl .(3.12)

Proof. We start with the upper bound in (3.12). Using (3.5), the triangle inequal-

ity yields g′l(γ) = ‖∇sl(γ)‖20;ωl≤ (‖∇s‖0;ωl

+ ‖∇(s− sl(γ))‖0)2. The upper bound

for g′l(γ) is now easy to see by noting that ‖∇s‖0;ωl≤ Ch

2π/θl and recalling (3.3). We

emphasize that C depends on k0 but not on hl.To show the lower bound in (3.12), we proceed in two steps. We first consider

g′′l (γ) and then g′l(0). The second derivative is given by g′′l (γ) = 2al(s′l(γ), sl(γ)).

Setting vl = s′l(γ) in (3.6), we get

(3.13) g′′l (γ) = 2aec(s′l(γ), s

′l(γ)) ≥ 0,

and thus g′l(γ) ≥ g′l(0) ≥ 0. The proof of a lower bound for g′l(0) is based on the alge-braic representation of gl(γ) and on g′l(γ). Using (3.8), (3.9), the fact that gl(γl) = 0,and the equality

(AII − γB)−1 = A−1II + γA−1

II B12 (Id− γB

12A−1

II B12 )−1B

12A−1

II , γ ∈ [0, 0.5],

we can bound −gl(0) in terms of g′l(0) by

−gl(0) = −a(s, s) + s�R(ARR −ARIA−1II AIR)sR

= −gl(γl) + γls�RARIA

−1II B

12 (Id− γlB

12A−1

II B12 )−1B

12A−1

II AIRsR

≤ γl‖(Id− γlB12A−1

II B12 )−1‖g′l(0)

≤ γl1− γl

g′l(0) ≤1

3g′l(0) ≤ g′l(0).

These preliminary considerations show in terms of (3.11) that there exists a level-independent constant such that the lower bound in (3.12) is satisfied.

Now we can set Jl := [γl − τh2(1−π/θ)l , γl + τh

2(1−π/θ)l ] ∩ [0, 0.5] with τ > 0 and

reasonably small. Then we find for γ ∈ Jl that |gl(γ)| ≤ τβh2l , and for γ ∈ Jl an

energy-corrected finite element method is obtained.

3.5. Newton algorithm. Our theoretical findings allow us to formulate a glob-ally convergent Newton algorithm.

Algorithm 1. Determine ωl and calculate γ on level l ≥ l0.

k0 := 1while gl(0.5) ≤ 0 dok0 ← k0 + 1

end whileσ := 0.01k := 0; γ0 := 0.5while gl(γ

k) > σh2l do

γk+1 := γk − a(s,s)−aec(sl(γk),sl(γ

k))al(sl(γk),sl(γk))

k ← k + 1end while

Each Newton step requires the solution of one finite element system (3.2) andsome extra O(1) cost evaluations since sl(γ)− sl(0.5) ∈ H1

0 (Ω). The parameter σ canbe configured for the problem at hand. Too small values may require more Newton


steps, while for larger values, the approximation of γl may be too poor. In all ourtests, setting σ = 0.01 worked very well.

Theorem 3.5. Algorithm 1 terminates and defines an energy-corrected finiteelement method.

Proof. The theoretical results of section 3.3 guarantee the existence of a finitelevel-independent k0 such that gl(0.5) > 0, and thus the first while statement termi-nates.

To show that the second while statement terminates, we first establish that thesequence gl(γ

k) is strictly decreasing. Since g′′l (γ) ≥ 0 (see (3.13)) and gl(0.5) > 0, allour γk satisfy 0 < γl ≤ γk ≤ 0.5 and thus gl(γ

k) ≥ 0. A straightforward computationshows in terms of the equivalence (3.12) that

gl(γk) ≤

(1− α

β

)gl(γ

k−1) ≤(1− α

β

)k

gl(0.5).

Since gl(0.5) ≤ Ch2π/θl , the number of required Newton steps is at most O(l).

Remark 3.6. For all our numerical results k0 is equal to one. If N + 1 is thenumber of elements in ωl, then only 3N − 2 entries of the stiffness matrix have to bemodified. Typically N = 2, 3, 4, 5, 6, 7, hence less than 20 entries have to be modified;see also Figure 2, where N = 4

Remark 3.7. A nested Newton variant improves the performance significantly.In particular, if gl(γ

0) = O(h2l ), then the number of required Newton steps will be

level-independent. This is the case, e.g., for γ0 := γl−1.

4. Convergence of the level-dependent correction factor. The results ofthe previous sections indicate that on each level we have to solve a nonlinear problemto determine an accurate enough approximation for γl. This can be done by Algo-rithm 1, where each step requires the solution of a finite element problem. Thus forhigher levels it is quite expensive and makes the scheme unattractive for practicalapplications. We are then interested in designing nested Newton schemes which canbe easily embedded in a full multigrid method and which require only one Newtonstep per refinement level. To guarantee global convergence for such a scheme we haveto provide theoretical results for the energy defect function.

4.1. Properties of the energy defect function. In this subsection, we con-sider in more detail the properties of gl(·) given by (3.4) as a function of γ. Recallthat, different from [19], we do not work with homogeneous Dirichlet boundary con-ditions but with a boundary value problem with a homogeneous right-hand side. Aswe will see, in this case gl(·) is strictly increasing.

Lemma 4.1. For the constants 0 < α ≤ β <∞ and l0 of Lemma 3.4 we have forl ≥ l0 and γ, γ ∈ [0, 0.5]

0 ≤ g′′l (γ) ≤2

1− γg′l(γ),(4.1a)

g′′l (γ) ≤4β

αg′l(γ).(4.1b)

Proof. The proof of (4.1a) is based on the equality in (3.13) and the upperbound (3.9). Denoting the algebraic representation of s′l(γ) with dsγ , then dsγR = 0.Moreover we have g′′l (γ) = 2(sγI )

�BdsγI , and (3.5) yields gl(γ) = (sγI )�BsγI . In terms


of (3.9), we then get

(1− γ)(dsγI )�BdsγI ≤ (dsγI )

�(AII − γB)dsγI = aec(s′l(γ), s

′l(γ))

= al(s′l(γ), sl(γ)) = (sγI )

�BdsγI ≤((sγI )

�BsγI) 1

2((dsγI )

�BdsγI) 1

2

and therefore (1− γ)((dsγI )

�BdsγI) 1

2 ≤((sγI )

�BsγI) 1

2 , i.e.,

g′′l (γ) ≤2

1− γ

((sγI )

�BsγI) 1

2((sγI )

�BsγI) 1

2 =2

1− γ(sγI )

�BsγI =2

1− γg′l(γ).

Finally, the proof of (4.1b) follows directly from (3.12) and (4.1a).

4.2. Convergence of γl. The choice γ = γl results in a fairly expensive algo-rithm if the γl are not precomputed. The main theoretical result of this section isto establish convergence of γl. This observation then allows us to formulate a nestedone-step Newton algorithm for the approximation of γl. To link γl with γl+1, we firstintroduce an auxiliary quantity γi

l , i = 1, 2, and second relate γil to γl.

Let Ωi ⊂ Ω be defined by Ωi := ω1i , i = 1, 2, 3; see Figure 3 for cases i = 1 and

i = 2. Associated with Ωi, we define discrete solutions sil(γ) ∈ Vl and sil(γ) ∈ Vl

such that both are given by interpolation on Ω \Ωi, i.e., sil(γ)(pl) = sil(γ)(pl) = s(pl)

for all vertices of Tl being in Ω \ Ωi and by variational equality on Ωi, and Ωi \ Ω3,respectively, i.e.,

a(sil(γ), vl)− γal(sil(γ), vl) = 0, vl ∈ Vl, v|Ωi ∈ H1

0 (Ωi),(4.2a)

a(sil(γ), vl) = 0, vl ∈ Vl, supp vl ⊂ Ωi \ Ω3.(4.2b)

To obtain a well-defined sil(γ) on Ω, we set it equal to sl(γ) on Ω3.

1Ω

Ω2

Fig. 3. The construction of Ωi, i = 1, 2.

Similarly to (3.2), (3.4), and γl, we define now γil , i = 1, 2, as the unique solution

of

(4.3) gil (γ) := a|Ωi(s, s)− a|Ωi(sil(γ), s

il(γ)) + γal(s

il(γ), s

il(γ)) = 0,

where the discrete solution sil(γ) ∈ Vl on Ωi is defined by (4.2a). Here we use thenotation a|X(·, ·) for the bilinear form a(·, ·) with the integral restricted to the sub-domain X ⊂ Ω. We point out that the results of section 3 also apply for the energydefect functions gil(·), i = 1, 2.

Lemma 4.2. Let γ1l and γ2

l+1 be the unique correction parameters on level l andl + 1 such that (4.3) holds for i = 1 and i = 2, respectively. Then we have

γ1l = γ2

l+1.


Proof. Without loss of generality, we have assumed that xc is the origin of thecoordinate system, and thus the linear mapping F (x) = 2x maps Ω2 onto Ω1. More-over for x ∈ ∂Ω2, we find that F (x) ∈ ∂Ω1 and s ◦ F = 2π/θs. By construction of Ω1

and Ω2, the boundary nodes on level l + 1 of domain Ω2 are mapped by F onto theboundary nodes on level l of domain Ω1, and ωl+1 is mapped onto ωl. The definition(4.2a) of sil(γ) now yields

s2l+1(γ)(x) = 2−π/θs1l (γ)(2x), x ∈ Ω2,

and thus in terms of (4.3), we get g1l (γ) = 22π/θg2l+1(γ).Lemma 4.3. Let γl and γi

l , i = 1, 2, be the correction parameters on level l suchthat gl(γl) = 0 and (4.3) hold, respectively. Then we have

|γil − γl| ≤ Ch

2(1−π/θ)l .

Proof. In a first step, we provide an upper bound for γl − γil . Using the fact that

sil(γil )− sl(γl) ∈ H1

0 (Ω), we find in terms of (3.2) that

a(sil(γil ), s

il(γ

il ))− γlal(s

il(γ

il ), s

il(γ

il )) ≥ a(sl(γl), sl(γl))− γlal(sl(γl), sl(γl)).

Now the definition of γl yields

gil := a(s, s)− a(sil(γil ), s

il(γ

il )) + γlal(s

il(γ

il ), s

il(γ

il )) ≤ gl(γl) = 0.

Decomposing Ω into Ωi and Ωoi := Ω \ Ωi, we can rewrite gil as

gil = a|Ωi(s, s)− a|Ωi(sil(γ

il ), s

il(γ

il )) + γi

lal(sil(γ

il ), s

il(γ

il ))

+ a|Ωoi(s, s)− a|Ωo

i(sil(γ

il ), s

il(γ

il )) + (γl − γi

l )al(sil(γ

il ), s

il(γ

il ))

= gil(γil ) + a|Ωo

i(s, s)− a|Ωo

i(Ils, Ils) + (γl − γi

l )al(sil(γ

il ), s

il(γ

il ))

= a|Ωoi(s, s)− a|Ωo

i(Ils, Ils) + (γl − γi

l )al(sil(γ

il ), s

il(γ

il )).

To obtain an upper bound for γl − γil , we recall that (gil )

′(γil ) = al(s

il(γ

il ), s

il(γ

il ))

and thus get in terms of Lemma 3.4 that al(sil(γ

il ), s

il(γ

il )) ≥ ch

2π/θl > 0. On the

other hand, noting that dist(x, xc) = O(1) for all x ∈ Ωoi , we find |a|Ωo

i(s, s) −

a|Ωoi(Ils, Ils)| ≤ Ch2

l . These two observations yield

(4.4) γl − γil ≤ Ch

2(1−π/θ)l .

The proof of the lower bound for γl − γil requires the use of sil(γ) and applies similar

arguments. We note that sil(γil )− sil(γl) restricted to Ωi is in H1

0 (Ωi), i = 1, 2. Thenthe definition (4.2a) yields that

a|Ωi(sil(γl), s

il(γl))− γi

lal(sil(γl), s

il(γl)) ≥ a(sil(γ

il ), s

il(γ

il ))− γi

lal(sil(γ

il ), s

il(γ

il )).

Now, using the definition of γil and the fact that gl(γl) = 0, we find in terms of

sil(γl) = sl(γl) on Ω3 and sil(γl) = Ils on Ωoi that

0 = gil(γil ) ≥ a|Ωi(s, s)− a|Ωi(s

il(γl), s

il(γl)) + γi

lal(sil(γl), s

il(γl))

= a(s, s)− a(sl(γl), sl(γl)) + γlal(sl(γl), sl(γl)) + a|Ωoi(Ils, Ils)− a|Ωo

i(s, s)

+ a(sl(γl), sl(γl))− a(sil(γl), sil(γl)) + (γi

l − γl)al(sl(γl), sl(γl))

= a|Ωoi(Ils, Ils)− a|Ωo

i(s, s)− ‖∇(sil(γl)− sl(γl))‖20 + (γi

l − γl)al(sl(γl), sl(γl)).


Applying the definition (4.2b), we get ‖∇(sil(γl)−sl(γl))‖20 = ‖∇(sil(γl)−sl(γl))‖20;Ωo3.

In a next step, we can further decompose Ωo3 into Ωo

i and Ωi \ Ω3. On Ωi \ Ω3, bothdiscrete functions sil(γl) and sl(γl) are discrete harmonic. Moreover, sil(γl)− sl(γl) isequal to zero on ∂Ω3. As a consequence, the H1-seminorm on Ωi \Ω3 can be bounded

by the H1/200 -seminorm on ∂Ωi ∩ Ω, which can be bounded by the H1-seminorm on

Ωoi ; see, e.g., [31]. Using sil(γl) = Ils on Ωo

i , we have due to (3.3)

‖∇(sil(γl)− sl(γl))‖0 ≤ C‖∇(sil(γl)− sl(γl))‖0;Ωoi

≤ C(‖∇(Ils− s)‖0;Ωo

i+ ‖∇(s− sl(γl))‖0;Ωo

i

)≤ Chl.

Finally, we recall that al(sl(γl), sl(γl)) ≥ ch2π/θl > 0, which implies

γil − γl ≤ Ch

2(1−π/θ)l .

Having the upper bound (4.4) and the lower bound, the distance between γil and γl

can be bounded.Theorem 4.4. Let ωl be such that the root γl ∈ [0, 0.5]. Then we have for

π < θ < 2π

|γl − γl+1| ≤ Ch2(1−π/θ)l .

Moreover γl converges to γ∞ ∈ (0, 0.5] with |γl − γ∞| ≤ Ch2(1−π/θ)l and

|gl(γ∞)| ≤ Ch2l .

Proof. Combining the results of Lemmas 4.2 and 4.3, we get

|γl − γl+1| = |γl − γ1l + γ2

l+1 − γl+1| ≤ C(h2(1−π/θ)l + h

2(1−π/θ)l+1

)≤ Ch

2(1−π/θ)l .

Using the triangle inequality and the boundedness of a geometric series, it can beeasily seen that γl defines a Cauchy sequence and converges with the given rate. Toprove the bound for gl(γ∞), we use the fact that there exists a ξl ∈ [0, 0.5] such that

|gl(γ∞)| = |gl(γl) + (γ∞ − γl)g′l(ξl)| ≤ Ch

2(1−π/θ)l h

2π/θl ,

which follows by Taylor expansion and Lemma 3.4.

5. Nested Newton algorithms. Although Algorithm 1 converges globally, itdoes not exploit the convergence of γl. In this section, we present several nestedNewton strategies for the approximative computation of γ∞. We assume that westart with a coarse initial mesh and use uniform refinement.

5.1. A nested Newton iteration using the exact energy: Algorithm 2.Our first algorithm is a simple Newton strategy based on the observation that γl isthe root of gl(·). We select γa

0 ∈ [0, 0.5] and for l = 0, 1, . . . compute

(Algorithm 2) γal+1 := min(0.5,max(0, γa

trial)), γatrial := γa

l −gl(γ

al )

g′l(γal )

.

Recall that the index l stands for the refinement level, and on each level only oneNewton step is carried out. There is no need for reiteration, and there is almost noextra computational cost, provided that we know a(s, s).

Theorem 5.1. Let us assume that ωl = ωk0

l with k0 fixed such that γ∞ < 0.5 ifk0 > 1 or γ∞ ≤ 0.5 if k0 = 1. Then there exists a constant Ca <∞ such that

(5.1) |γal+1 − γ∞| ≤ C(q, γ∞)|γa

l − γ∞|2 + Cah2(1−π/θ)l ,


with C(q, γ∞) := q1−qγ∞

and q as in (3.9). Thus we have convergence for all γa0 ∈

[0, 0.5].Proof. Let C1 := AII−γ1B and C2 := AII−γ2B; then we have for γ1, γ2 ∈ [0, 0.5]

the relation

C−11 = C−1

2 + (γ1 − γ2)C−12 B

12 (Id− (γ1 − γ2)B

12C−1

2 B12 )−1B

12C−1

2

= C−12 + (γ1 − γ2)C

−12 BC−1

2

+(γ1 − γ2)2C−1

2 BC− 1

22

( ∞∑l=0

(γ1 − γ2)l(C

− 12

2 BC− 1

22 )l

)C

− 12

2 BC−12

= C−12 + (γ1 − γ2)C

−12 BC−1

2

+(γ1 − γ2)2C−1

2 BC− 1

22

(Id− (γ1 − γ2)(C

− 12

2 BC− 1

22 )

)−1

C− 1

22 BC−1

2

= C−12 + (γ1 − γ2)C

−12 BC−1

2 + (γ1 − γ2)2C−1

2 B(C2 − (γ1 − γ2)

lB)−1

BC−12

= C−12 + (γ1 − γ2)C

−12 BC−1

2 + (γ1 − γ2)2C−1

2 BC−11 BC−1

2 .

In terms of this elementary equality and by means of (3.8a) and (3.8b), we obtain

gl(γ1) = gl(γ2) + (γ1 − γ2)g′l(γ2) + (γ1 − γ2)

2s�RARIC−12 BC−1

1 BC−12 AIRsR,(5.2)

gl(γ1) ≤ gl(γ2) + (γ1 − γ2)g′l(γ2) + (γ1 − γ2)

2‖B 12C−1

1 B12 ‖g′l(γ2).

In contrast to a standard Taylor expansion, the quadratic term in (γ2−γ1)2 is weightedby the first derivative and not by a second one. Setting γ1 = γ∞, γ2 = γa

l and using(3.8b), we find

(5.3)gl(γ∞)− gl(γ

al )

g′l(γal )

= (γ∞ − γal ) + (γ∞ − γa

l )2 s

�RARIC

−12 BC−1

1 BC−12 AIRsR

s�RARIC−12 BC−1

2 AIRsR.

The definition of γatrial and (5.3) result in

|γal − γ∞| ≤ |γa

trial − γ∞| =∣∣∣∣γa

l − γ∞ −gl(γ

al )− gl(γ∞)

g′l(γal )

− gl(γ∞)

g′l(γal )

∣∣∣∣≤ (γa

l − γ∞)2‖B 12 (AII − γ∞B)−1B

12 ‖+

∣∣∣∣gl(γ∞)

g′l(γal )

∣∣∣∣ .Due to (3.9) the first term on the right can be bounded, and Theorem 4.4 and Lemma3.4 yield a bound for the second term. Altogether we get

|γal+1 − γ∞| ≤ |γa

trial − γ∞| ≤q

1− qγ∞(γa

l − γ∞)2 + Ch2l h

−2π/θl

≤ C(q, γ∞)max(γ∞, 0.5− γ∞)|γal − γ∞|+ Ch2

l h−2π/θl .

Under the assumptions on γ∞ and k0, we have that the first term on the right is acontraction, and global convergence is obtained.

5.2. A nested Newton iteration on two levels: Algorithm 3. The maindisadvantage of Algorithm 2 is that the determination of γa

l+1 requires the exact eval-uation of gl(γ

al ), which depends on a(s, s) and sl(γ

al ). The unknown energy a(s, s)

can possibly be evaluated analytically or up to order h2l accurate by quadrature for-

mulas for∫∂Ω

s · ∂ns ds on the edges of ∂Ω. Our main interest is the formulation of


an algorithm which does not require this evaluation. A first step into this directionis to propose an algorithm which does not require the explicit knowledge of a(s, s).

To this end we define an alternative characterization of the correction parame-ter by requiring approximately that the energy defect function on two consecutivelevels coincide: gl(γ) = gl−1(γ). We then set γb

0 = γb1 = 0 and define γb

l+1 :=

min(0.5,max(0, γbtrial)), l = 1, 2, . . . , with

(Algorithm 3) γbtrial :=

a(sl(γbl ), sl(γ

bl ))− a(sl−1(γ

bl ), sl−1(γ

bl ))

al(sl(γbl ), sl(γ

bl ))− al−1(sl−1(γb

l ), sl−1(γbl ))

.

Algorithm 3 is motivated by the observation that γbtrial can be equivalently written as

(5.4) γbtrial = γb

l −gl(γ

bl )− gl−1(γ

bl )

g′l(γbl )− g′l−1(γ

bl )

.

This can be interpreted as one Newton step with start iterate γbl applied for solving

gl(γ) = gl−1(γ).Before we consider the convergence of the sequence γb

l given by Algorithm 3, wedevelop a relation between gl(γ) and gl+1(γ).

Lemma 5.2. The energy defect function on level l − 1 is related to the energydefect function on level l by

gl−1(γ) = 22π/θgl(γ) +O(h2l ),(5.5a)

g′l−1(γ) = 22π/θg′l(γ) +O(h1+π/θl ).(5.5b)

Proof. We first use the results and notation of section 4 to relate gl(γ) to g2l (γ)and gl−1(γ) to g1l−1(γ) via

gl(γ) = g2l (γ) + (a|Ω◦2(s, s)− a|Ω◦

2(Ils, Ils)) + ‖sl(γ)− s2l (γ)‖2hl;γ

,

gl−1(γ) = g1l−1(γ) + (a|Ω◦1(s, s)− a|Ω◦

1(Ils, Ils)) + ‖sl−1(γ)− s1l−1(γ)‖2hl−1;γ ,

where we recall that ‖vl‖2hl;γ:= a(vl, vl) − γal(vl, vl). Using the same arguments as

in the proof of Lemma 4.3, we find that the second and third terms on the right areof order h2

l . Taking into account the equality g1l−1(γ) = 22π/θg2l (γ) yields (5.5a).To obtain a similar relation for the derivatives, we have to consider g′l(γ), g

′l−1(γ)

and (g2l )′(γ), (g1l−1)

′(γ) in more detail. Starting with the trivial equalities

g′l(γ)− (g2l )′(γ) = al(sl(γ), sl(γ))− al(s

2l (γ), s

2l (γ)),

g′l−1(γ)− (g1l−1)′(γ) = al−1(sl−1(γ), sl−1(γ))− al−1(s

1l−1(γ), s

1l−1(γ)),

we have to show that the differences on the right are of order h1+π/θl . Without loss of

generality, we restrict ourselves to the term Dl := |al(sl(γ), sl(γ))− al(s2l (γ), s

2l (γ))|.

Using the definitions of sl(γ) and s2l (γ), we get

Dl ≤ al(sl(γ)− s2l (γ), sl(γ)− s2l (γ)) + 2|a(sl(γ), sl(γ)− s2l (γ))|

≤ Ch2l + 2|al(sl(γ), sl(γ)− s2l (γ))| ≤ C

(h2l + h

π/θl hl

)≤ Ch

1+π/θl ,

and thus (5.5b) holds.Theorem 5.3. Let us assume that ωl = ωk0

l with k0 fixed such that γ∞ < 0.5 ifk0 > 1 or γ∞ ≤ 0.5 if k0 = 1. Then there exists a constant Cb <∞ such that

(5.6) |γbl+1 − γ∞| ≤ C(q, γ∞)|γb

l − γ∞|2 + Cbh2(1−π/θ)l ,

with C(q, γ∞) as in Theorem 5.1. Thus we have convergence for all γb0 ∈ [0, 0.5].


Proof. The proof is based on the equivalent representation of γbtrial by (5.4). We

follow along the lines of the proof of Theorem 5.1, use (5.5a) and (5.5b), and get forhl small enough

|γbtrial − γ∞| =

∣∣∣∣∣γbl − γ∞ −

gl(γbl )− gl−1(γ

bl )

g′l(γbl )− g′l−1(γ

bl )

∣∣∣∣∣=

∣∣∣∣∣γbl − γ∞ +

(22π/θ − 1)gl(γbl ) +O(h2

l )

(22π/θ − 1)g′l(γbl ) +O(h

1+π/θl )

∣∣∣∣∣≤∣∣∣∣γb

l − γ∞ +gl(γ

bl )

g′l(γbl )

∣∣∣∣+ C(h2l h

−2π/θl + gl(γ

bl )h

1+π/θl h

−2π/θl

)≤ C(q, γ∞)(γb

l − γ∞)2 + Ch1−π/θl

(h1−π/θl + h2

l + |γbl − γ∞|h2π/θ

l

)≤ C(q, γ∞)(γb

l − γ∞)2 + Cbh2(1−π/θ)l .

We note that in the last step, we have used 1 − π/θ ≤ 2π/θ and that |γbl − γ∞|

is bounded. Let (5.6) hold; then the first term on the right is by construction acontraction, and thus global convergence can be observed.

5.3. An inexact nested Newton iteration: Algorithm 4. Although Algo-rithm 3 does not require the value a(s, s), we have to solve the finite element equationon both levels, l and l − 1, with the given value γb

l . Thus we propose a furthersimplification of the algorithm where we reuse the results of the previous computationsby replacing γb

l in the finite element approximation on level l− 1 by γbl−1.

We set γc0 = γc

1 = 0 and define γcl+1 := min(0.5,max(0, γc

trial)), l = 1, 2, . . . , with

(Algorithm 4) γctrial :=

a(sl(γcl ), sl(γ

cl ))− a(sl−1(γ

cl−1), sl−1(γ

cl−1))

al(sl(γcl ), sl(γ

cl ))− al−1(sl−1(γc

l−1), sl−1(γcl−1))

.

Theorem 5.4. Under the assumptions of Theorem 5.3, there exist a τ > 0 smallenough and a level l0 such that for |γc

l0−γ∞|, |γc

l0+1−γ∞| ≤ τ , we have |γcl −γ∞| ≤ τ ,

l ≥ l0, and moreover

(5.7) |γcl+1 − γ∞| ≤ Cc

(h2(1−π/θ)l + |γc

l − γ∞|2 + |γcl−1 − γ∞|2

), l ≥ l0,

with Cc <∞, and thus local convergence is guaranteed.Proof. The proof is technical but essentially follows the arguments of the proofs of

Theorems 5.1 and 5.3. Thus we do not work out all details on the constants but onlysketch the main aspects. We start by reformulating the definition of Algorithm 4. Us-ing (3.4) and (3.5), we can rewrite γc

trial in terms of gl−1(γcl−1), gl(γ

cl ) and g′l−1(γ

cl−1),

g′l(γcl ), obtaining

γctrial =

gl−1(γcl−1)− γc

l−1g′l−1(γ

cl−1)− gl(γ

cl ) + γc

l g′l(γ

cl )

g′l(γcl )− g′l−1(γ

cl−1)

= γcl +

gl−1(γcl−1)− gl(γ

cl ) + (γc

l − γcl−1)g

′l−1(γ

cl−1)


cl−1)

.

Using the notation of the proof of Theorem 5.1 on level l−1, and setting γ1 = γcl , γ2 =

γcl−1 in (5.2), we can reformulate the numerator as

γctrial = γc

l −gl(γ

cl )− gl−1(γ

cl ) + (γc

l − γcl−1)

2s�RARIC−12 BC−1

1 BC−12 AIRsR


cl−1)

.


Taylor expansion of g′l(·) around γcl−1 with a suitable ξl ∈ [0, 0.5 + τ ] yields in terms

of (5.5b) that

g′l(γcl ) = g′l(γ

cl−1) + (γc

l − γcl−1)g

′′l (ξl)

= 2−2π/θg′l−1(γcl−1) +O(h

1+π/θl ) + (γc

l − γcl−1)g

′′l (ξl).

Now due to (3.12) and (4.1b), we get for a suitable σ <∞ independent of l

|g′l(γcl )− g′l−1(γ

cl−1)| = |(2−2π/θ − 1)g′l−1(γ

cl−1) +O(h

1+π/θl ) + (γc

l − γcl−1)g

′′l (ξl)|

≥ (1− 2−2π/θ − σh1−π/θl − σ|γc

l − γcl−1|)g′l−1(γ

cl−1).

For l ≥ l0, l0 large enough, and for |γcl − γc

l−1| ≤ 2τ , τ small enough, we find |g′l(γcl )−

g′l−1(γcl−1)| ≥ σg′l−1(γ

cl−1) with σ depending on l0, τ, and θ but not on l ≥ l0. We recall

that for γcl ≤ 3/5, we have B1/2C−1

1 B1/2 ≤ B1/2(AII − 3/5B)−1B1/2 ≤ A1/2II (AII −

3/5AII)−1A

1/2II = 5/2 Id. These preliminary considerations yield by means of the

algebraic representation (3.8b)


1 BC−12 AIRsR

|g′l(γcl )− g′l−1(γ

cl−1)|

≤ 1

σ


1 BC−12 AIRsR

g′l−1(γcl−1)

=1

σ


1 BC−12 AIRsR


2 AIRsR≤ 5

2σ.

Using the triangle inequality, we obtain the upper bound

(5.8) |γctrial − γ∞| ≤

∣∣∣∣γcl − γ∞ −

gl(γcl )− gl−1(γ

cl )


cl−1)

∣∣∣∣+ 5

2σ(γc

l − γcl−1)

2.

To further bound the first term on the right, we follow along the lines of the proofof Theorem 5.1 and use gl(γ

cl ) − gl−1(γ

cl ) = (22π/θ − 1)gl(γ

cl ) + O(h2

l ) as well as

g′l(γcl ) − g′l−1(γ

cl−1) = (22π/θ − 1)g′l(γ

cl ) + O(h

1+π/θl ) + g′l−1(γ

cl ) − g′l−1(γ

cl−1). For

|γcl − γc

l−1| ≤ 2τ and τ small enough, we obtain by means of the properties of theenergy defect function∣∣∣∣γc


cl )− gl−1(γ

cl )


cl−1)

∣∣∣∣=

∣∣∣∣∣γcl − γ∞ −

(22π/θ − 1)gl(γcl ) +O(h2

l )

(22π/θ − 1)g′l(γcl ) +O(h

1+π/θl ) + (γc

l − γcl−1)g

′′l−1(ξl)

∣∣∣∣∣≤∣∣∣∣∣γc

l − γ∞ −(22π/θ − 1)gl(γ

cl )

(22π/θ − 1)g′l(γcl ) +O(h

1+π/θl ) + (γc

l − γcl−1)g

′′l−1(ξl)

∣∣∣∣∣ + Ch2(1−π/θ)l

≤∣∣∣∣γc


cl )

g′l(γcl )

∣∣∣∣+ C(|gl(γc

l )|(h1−π/θl + |γc

l − γcl−1|) + h

2(1−π/θ)l

)≤ C(γc

l − γcl−1)

2 + C(|γc

l − γ∞|(h1+π/θl + |γc

l − γcl−1|) + h

2(1−π/θ)l

).

Finally, (5.8) yields

|γcl+1 − γ∞| ≤ Cc

(h2(1−π/θ)l + |γc

l − γ∞|2 + |γcl−1 − γ∞|2

),


provided that the assumptions are satisfied, τ is small enough, and l0 is large enough.Moreover if l0 is large enough, we obtain |γc

l+1 − γ∞| ≤ τ , and thus (5.7) follows byinduction.

Remark 5.5. In contrast to Theorems 5.1 and 5.3, Theorem 5.4 does not guaranteeglobal but only local convergence. Although all three upper bounds (5.1), (5.6), and(5.7) have the same structure, there is one characteristic difference. In (5.1) and (5.6)the constants in front of the quadratic error terms can be more precisely specified, andthus convergence for all admissible start iterates is given. Starting with Algorithm 3on coarse levels and then switching to Algorithm 4 guarantees global convergence.

Remark 5.6. Although Algorithms 3 and 4 can be applied without the explicitknowledge of a(s, s), we still require s to set the boundary conditions of our auxiliaryproblems. However, in any problem setup where the singular component s causes thedominating error contribution, the same algorithms can be used.

6. Numerical results. Unless mentioned otherwise, we consider the problem(3.1) where we know the exact solution that is given by the singular function u = s.Starting from coarse meshes T0, we generate a sequence of meshes Tl by uniform mid-point refinements. For our numerical tests, we often consider the symmetric (circular)L-shape geometry and the slit domain with interior angles θ = 3π/2 and θ = 2π, re-spectively; cf. Figure 4.

Fig. 4. Meshes T0 and T1 for n = 6 elements attached to the re-entrant corner and interiorangles θ = 3π/2 and θ = 2π, respectively.

All implementations are based on the Python interface of the DOLFIN (v. 1.2.0)finite element environment [24].

6.1. A comparison of the nested Newton algorithms. In the following nu-merical experiments, we compute the errors in weighted L2-norms defined in (2.4).For the standard finite element method without energy correction, we expect a sub-optimal asymptotic convergence rate of 2π/θ, i.e., 4/3 for the L-shape and 1 for theslit domain. This behavior can indeed be observed in Tables 4 and 5 for the series offinite element solutions obtained without energy correction; see also [19].

Next, we compare the different variants of the energy corrected method, as pro-posed in sections 3 and 5, namely, the exact Newton on each level (Algorithm 1), thenested iteration in terms of the exact energy (Algorithm 2), the approach on two levels(Algorithm 3), and the inexact method (Algorithm 4). The errors for subsequentlyrefined meshes of the L-shaped and slit domains are displayed in Tables 4 and 5,respectively. All four algorithms successfully recover the optimal asymptotic conver-gence rates. Furthermore, in terms of the absolute error, the three nested algorithmsreach almost the same or even slightly better results than when using Algorithm 1 toapproximate the root γl. The underlined values indicate that with energy correctionapproximately two levels of refinement can be saved in case of the L-shaped domain,as compared with the unmodified finite element solution. In the case of the slit do-main, the error on level 3 with correction is already smaller than the error on level 6for the uncorrected solution.


Table 4

L2ρ errors obtained with uncorrected vs. energy corrected methods for n = 6 and θ = 3π/2.

Uncorrected Algorithm 1 Algorithm 2 Algorithm 3 Algorithm 4

l Error Rate Error Rate Error Rate Error Rate Error Rate

0 2.362e-2 - 1.774e-2 - 1.696e-2 - 1.706e-2 - 1.706e-2 -

1 8.248e-3 1.52 4.458e-3 1.99 4.272e-3 1.99 4.283e-3 1.99 4.105e-3 2.06

2 3.002e-3 1.46 1.115e-3 2.00 1.068e-3 2.00 1.070e-3 2.00 1.071e-3 1.94

3 1.129e-3 1.41 2.788e-4 2.00 2.672e-4 2.00 2.678e-4 2.00 2.676e-4 2.00

4 4.337e-4 1.38 6.973e-5 2.00 6.686e-5 2.00 6.701e-5 2.00 6.701e-5 2.00

5 1.688e-4 1.36 1.743e-5 2.00 1.673e-5 2.00 1.676e-5 2.00 1.676e-5 2.00

6 6.621e-5 1.35 4.361e-6 2.00 4.186e-6 2.00 4.195e-6 2.00 4.195e-6 2.00

Table 5

L2ρ errors obtained with uncorrected vs. energy corrected methods for n = 6 and θ = 2π.

Uncorrected Algorithm 1 Algorithm 2 Algorithm 3 Algorithm 4

l Error Rate Error Rate Error Rate Error Rate Error Rate

0 4.245e-2 - 2.587e-2 - 2.456e-2 - 2.472e-2 - 2.472e-2 -

1 1.874e-2 1.18 6.397e-3 2.02 6.251e-3 1.97 6.234e-3 1.99 8.819e-3 1.49

2 8.890e-3 1.08 1.587e-3 2.01 1.561e-3 2.00 1.557e-3 2.00 1.605e-3 2.46

3 4.347e-3 1.03 3.945e-4 2.01 3.892e-4 2.00 3.887e-4 2.00 4.351e-4 1.88

4 2.152e-3 1.01 9.817e-5 2.01 9.702e-5 2.00 9.692e-05 2.00 9.695e-5 2.17

5 1.071e-3 1.01 2.444e-5 2.01 2.417e-5 2.00 2.415e-05 2.00 2.415e-5 2.01

6 5.343e-4 1.00 6.088e-6 2.01 6.024e-6 2.00 6.019e-6 2.00 6.019e-6 2.00

In Figure 5 we plot the evolution of the correction parameters γl with the re-finement levels l for the different algorithms. We point out that all three nestedNewton algorithms proposed in section 5 converge with the same order but quantita-tively faster to γ∞ than to the mesh-dependent root γl of the energy defect function.Algorithms 2 and 3 produce almost the same curves, whereas the cheapest method,Algorithm 4, shows oscillations in the preasymptotic range; cf. also Remark 5.5.

0 2 4 6 8 10 120.100

0.105

0.110

0.115

0.120

0.125

0.130

Algorithm 1

Algorithm 2

Algorithm 3

Algorithm 4

0 2 4 6 8 10 120.24

0.26

0.28

0.30

0.32

0.34

0.36

0.38

Algorithm 1

Algorithm 2

Algorithm 3

Algorithm 4

Fig. 5. Plot of γl obtained by the different nested Newton methods, depending on level l for

θ = 3π/2 (left) and θ = 2π (right). The dashed lines indicate nonlinear fits to γl = c0 + c1h2−2π/θl ,

starting from levels 2 and 4, respectively, to exclude preasymptotical effects.


6.2. The influence of the domain and the coarse mesh on γ∞. In thissubsection, we consider numerically the influence of the interior angle θ at the re-entrant corner and of the number of elements touching the re-entrant corners on γ∞.All tests are set up with k0 = 1, and thus ωl is the union of the n elements T ∈ Tlsuch that xc is a vertex of T . The asymptotic parameter γn

∞ = γn∞(θ) is regarded as

a function on θ ∈ [π, 2π]. Here isosceles triangles with angle θ/n are used; the effectsof the element shapes and asymmetry near the corner are discussed in [19].

To obtain an approximation of γ∞ that is sufficiently accurate for many practicalcomputations and to save an explicit computation, we propose a nonlinear fit whichis constructed as follows: For each n ∈ {3, . . . , 12} and 60 samples of θ in [π, 2π],we apply Algorithm 3 to compute parameters γl on a series of 7 meshes, which aresuccessively refined according to the Bulirsch sequence (i.e., 1, 1

2 ,13 ,

14 ,

16 , . . . ) in

order to reduce the cost per evaluation; cf. Figure 6.

Fig. 6. Series of meshes refined according to the Bulirsch sequence for n = 6 and θ = 7π/4.

Given this data, and assuming an asymptotic expansion for γ(h) = γ∞+c1h2−2π/θ+

o(h2−2π/θ), we use a Richardson extrapolation of the form

(6.1) γ∞(θ, n) ≈ γl + (γl+1 − γl)/(1−(hl+1

hl

)2−2π/θ)

on the last level to eliminate the dominating error term in the asymptotic expansionand obtain improved values for the correction parameters. Having a closer look atthe numerical results obtained by this procedure, we find furthermore that γ∞(θ, n)can be accurately approximated by a fit in the form of

γfit(θ, n) = σ1,n(exp(−2(θ − π))− 1) + σ2,n(θ − π).

In Table 6, we list the coefficients σ1,n and σ2,n obtained for Dirichlet- and Neumann-type singularities for typical numbers of elements attached to the singularity.

Table 6

Coefficients obtained from the nonlinear fit.

Dirichlet corner Neumann corner

n σ1,n σ2,n σ1,n σ2,n

3 0.0998183980437 0.1896155427030 0.1033975735530 0.1917937440240

4 0.0555624819392 0.1280415576990 0.0559892102051 0.1282280329260

5 0.0415019850858 0.1072128902000 0.0425309204472 0.1076059179200

6 0.0363481781425 0.0979881012415 0.0369403570671 0.0981918095083

7 0.0328888599638 0.0925971779024 0.0319701432127 0.0922167915794

8 0.0313092655216 0.0894450110842 0.0303097852573 0.0890798605638

9 0.0304135897967 0.0874557266743 0.0296697187321 0.0871982940151

10 0.0289942470411 0.0857622163158 0.0290198736762 0.0858256530158

11 0.0279010670390 0.0844853256090 0.0273173027417 0.0842839957719

12 0.0279439846719 0.0838604929991 0.0268036813107 0.0834159415925


Numerical tests indicate that the straightforward implementation of this fit ofthe correction parameters in a finite element code already significantly improves thesolution in the presence of corner singularities; cf., e.g., [21] for an application toeigenvalue problems in nonconvex domains. Moreover, in convergence studies withrandomly chosen values of n and θ, we have always observed a substantially better so-lution compared to finite elements without energy correction. In addition, the fit alsoprovides a good initial guess for the previously discussed nested Newton algorithms.

Figure 7 illustrates the influence of n and θ on γ∞(θ, n). As the right plot indi-cates, the correction parameter assumes a maximal value at γ∞(2π, 3) = 1/2, and inthe numerical algorithms this value is obtained sharply. Whether this is mere chanceor not needs to be further investigated. Moreover we see that for n → ∞, γ∞(θ, n)converges and that for θ ∈ [3π/2, 2π], we obtain almost linear dependence for all n.

1.0 1.2 1.4 1.6 1.8 2.0

θ/π

0.0

0.1

0.2

0.3

0.4

0.5

γ∞

n=3.0

n=4.0

n=5.0

n=6.0

n=7.0

Fig. 7. Plot of γ∞ over θ and n.

6.3. An example with idealized cracks. Finally, we consider the domainΩ = [−2, 2] × [−1, 1] that has seven idealized cracks, i.e., re-entrant corners withθ = 2π, as depicted in Figure 8 (left). The problem is defined as

−Δu = 0 in Ω,

u = 14 cos(πx2) + 1 on Γ1 := {−2} × (−1, 1),

u = 14 cos(πx2) on Γ2 := {2} × (−1, 1),

∇u · n = 0 on ∂Ω\(Γ1 ∪ Γ2).

Such problems may arise, for example, in heat conduction studies in materials withcracks.

Given a triangulation of the domain, our implementation automatically detectsall corners and extracts the patch of elements attached to the corner. This time weuse the cheaper method, Algorithm 4, to compute a series of correction parameterson meshes, which are again successively refined according to the Bulirsch sequence.Using the Richardson extrapolation (6.1), we obtain sufficiently accurate values forthe correction parameters γ∞ = 0.28033007 . . . at the respective corners.

To measure the errors in this example, we use a weighted L2-norm

‖u− uh‖0,ρ = ‖ρ(u− uh)‖0,


Fig. 8. Left: Initial mesh and weighting function. Right: Plot of the solution on mesh level 3.

with the weighting function ρ defined as ρ := mini{2 |x−xi|1/2}, and where xi denotethe positions of the crack-tips, respectively. Figure 8 (left) illustrates the geometryand the weighting function. In addition to the weighted L2 error, we tabulate thediscretization error of the flux at the right boundary in the norm

‖h1/2∇(Πlu− ul)·n‖L2(Γ2),

where Πl denotes the L2 projection into the discrete function space Vl in which wecompute ul. Since we have no analytic solution u for this problem available, it isapproximated by the corrected method with mesh level 7. The convergence of theuncorrected vs. corrected finite element method is then examined on a series of uni-formly refined meshes. The results are shown in Tables 7 and 8. Additionally, thesolution on level 3 is depicted in Figure 8 (right).

Table 7

Numerical results for standard finite elements in the domain with cracks.

l L2 error Rate L2ρ error Rate Flux error Rate

0 2.176e-2 - 1.933e-2 - 1.463e-2 -

1 1.044e-2 1.06 9.013e-3 1.10 4.385e-3 1.74

2 5.112e-3 1.03 4.379e-3 1.04 1.414e-3 1.63

3 2.525e-3 1.02 2.163e-3 1.02 4.786e-4 1.56

4 1.254e-3 1.01 1.076e-3 1.01 1.660e-4 1.53

5 6.246e-4 1.01 5.373e-4 1.00 5.814e-5 1.51

Table 8

Numerical results for the energy corrected method in the domain with cracks.

l L2 error Rate L2ρ error Rate Flux error Rate

0 1.307e-2 - 1.078e-2 - 5.782e-3 -

1 4.032e-3 1.70 2.701e-3 2.00 1.253e-3 2.21

2 1.312e-3 1.62 6.784e-4 1.99 2.465e-4 2.35

3 4.418e-4 1.57 1.700e-4 2.00 4.598e-5 2.42

4 1.517e-4 1.54 4.267e-5 1.99 8.192e-6 2.49

5 5.260e-5 1.53 1.019e-5 2.06 1.194e-6 2.78

We again observe a beneficial effect of the energy correction, since the standardelement method exhibits reduced convergence, while optimal orders of convergenceare recovered by the correction. Additionally, the numerical accuracy of the fluxvalues can be significantly improved. Here, the discretization error in the uncorrected


solution is already of the theoretically optimal order O(h3/2). Surprisingly, the errorfor the energy-corrected approach yields a superconvergence effect, showing a rate ofroughlyO(h5/2) that cannot be predicted by the refined analysis in [27]. We conductedan additional set of tests with an uncorrected reference solution to ensure that thissuperconvergence is not due to the comparison with a solution obtained by the energycorrected method. However, the differences between both ways of evaluating errorswere negligible for both cases.

7. Conclusions and outlook. In this paper we have developed efficient algo-rithms and heuristics to determine the correction parameter that is required in theenergy correction technique for handling corner singularities in elliptic partial differ-ential equations. The algorithms are based on Newton’s method and can be embeddedefficiently in a multilevel context, where just a few or even only a single iteration is re-quired on each refinement level. Furthermore, we have demonstrated numerically thatthe coefficients can be tabulated and can be approximated by a heuristic function.Additionally, the theoretically shown asymptotic behavior of the correction parametercan be exploited to find improved values by performing Richardson extrapolation onthese values. In all these cases, optimal convergence in the L2-norm is recovered awayfrom the singularity.

Future work will deal with extensions of the energy correction technique, suchas second order finite elements, the computation of eigenvalues, and strongly het-erogeneous materials; see [21] for preliminary results. Finally, important directionsfor future research include a full integration in a multigrid solver framework and thepractical application to mechanical problems.

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