Numerical Methods for Maxwell Equations Joachim Sch¨ oberl April 8, 2009 Abstract The Maxwell equations describe the interaction of electric and magnetic fields. Important applications are electric machines such as transformers or motors, or elec- tromagnetic waves radiated from antennas or transmitted in optical fibres. To com- pute the solutions of real life problems on complicated geometries, numerical methods are required. In this lecture we formulate the Maxwell equations, and discuss the finite element method to solve them. Involved topics are partial differential equations, variational formulations, edge elements, high order elements, preconditioning, a posteriori error estimates. 1 Maxwell Equations In this chapter we formulate the Maxwell equations. 1.1 The equations of the magnetic fields The involved field quantities are B Vs m 2 magnetic flux density (germ: Induktion) H A m magnetic field intensity (germ: magn. Feldst¨ arke) j tot A m 2 electric current density (germ: elektrische Stromdichte) We state the magnetic equations in integral form. The magnetic flux density has no sources, i.e., for any volume V there holds Z ∂V B · n ds =0 Ampere’s law gives a relations between the magnetic field and the electric current. A current through a wire generates a magnetic field around it. For any surface S in space there holds: Z ∂S H · τ ds = Z S j tot · n ds 1
Numerical Methods for Maxwell Equations
Jun 21, 2023
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