An explicit P1 finite element scheme for Maxwell's equations with constant permittivity in a boundary neighborhood

Abstract

This paper is devoted to the complete convergence study of the approximation of Maxwell's equations in terms of the sole electric field, by means of standard linear finite elements for the space discretization, combined with a well-known explicit finite-difference scheme for the time discretization. The analysis applies to the particular case where the electric permittivity has a constant value outside a sub-domain, whose closure does not intersect the boundary of the domain where the problem is defined. Optimal convergence results are derived in natural norms under reasonable assumptions, provided a classical CFL condition holds.

Stability and convergence analysis of a domain decomposition FE/FD method for the Maxwell's equations in time domain

“…In [5,6], a finite element analysis shows stability and consistency of the stabilized finite element method for the solution of (1) with σ (x) = 0. In the current study we show stability and convergence for the combined FEM/FDM scheme, under the condition (3) on ε, where the stabilized FEM is used for the numerical solution of (2) in Ω FEM and usual FDM discretization of ( 5) is applied in Ω FDM .…”

“…In this section we construct, combined, finite element-finite difference schemes to solve the model problem (6). To do this, first we present the finite element scheme to solve (6) in entire Ω. This induces the finite element scheme in Ω FEM .…”

“…This induces the finite element scheme in Ω FEM . Then, we derive the finite difference scheme in Ω FDM , when the domain decomposition FE/FD structure is applied to solve (6) in Ω.…”

“…The computational schemes derived in this section are explicit and therefore, for their converegence, the CFL condition below (see ,e.g. [5,6]) should be satisfied…”

“…First we derive finite element scheme to solve the model problem (6) in whole Ω. Next, we discretize Ω FEM T = Ω FEM × (0, T ) in two steps: (i) the spatial discretization using a partition K h = {K} of Ω FEM into elements K, where h = h(x) is a mesh function defined as h| K = h K , representing the local diameter of elements.…”