Abstract
New inf-sup stable mixed elements are proposed and analyzed for solving the Maxwell equations in terms of electric field and Lagrange multiplier. Nodal-continuous Lagrange elements of any order on simplexes in two- and three-dimensional spaces can be used for the electric field. The multiplier is compatibly approximated always by the discontinuous piecewise constant elements. A general theory of stability and error estimates is developed; when applied to the eigenvalue problem, we show that the proposed mixed elements provide spectral-correct, spurious-free approximations. Essentially optimal error bounds (only up to an arbitrarily small constant) are obtained for eigenvalues and for both singular and smooth solutions. Numerical experiments are performed to illustrate the theoretical results.
Highlights
Let Ω ⊂ Rd, d = 2, 3, be a connected polygonal or polyhedral domain with connected Lipschitz-continuous boundary ∂Ω
This paper is concerned with the Lagrange elements for solving the following eigenproblem of Maxwell equations: (1.1)
To develop the finite element method (FEM), in addition to the electric field u, we introduce an additional new unknown variable p, which is called the Lagrange multiplier, in order to relax the div equation (Gauss law) so that it can hold in some weak other than pointwise sense
Summary
Since p = 0, the best approximation, denoted by some qh ∈ Qh, can be chosen as qh = 0, and the bilinear form b(·, ·) does not have any influence on the convergence of uh ∈ Kh. from the classical theory of saddle-point problems [12, Chapter II, section II.2.2], we can obtain the conclusions as stated in the above theorem, and the proof is omitted here.
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