A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients

Abstract

We perform the a posteriori error analysis of residual type of a transmission problem with sign changing coefficients. According to [6] if the contrast is large enough, the continuous problem can be transformed into a coercive one. We further show that a similar property holds for the discrete problem for any regular meshes, extending the framework from [6]. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.Key Words A posteriori estimator, non positive definite diffusion problems.

T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients

“…They rely either on the use of special meshes, or on the introduction of some dissipation, which amounts to adding some well-chosen imaginary number to σ. We finally devote our attention to the range of applicability of those discrete approaches, thus complementing the results of [3,20]. In the process, we provide error estimates, which we observe numerically on some examples.…”

“…The question to be addressed is how to build a family (T h ) h of discrete operators such that the form a is uniformly T h -coercive, at least for h small enough. Some methods have already been proposed in [3] and [20]. The first one relies on a lifting of the trace on the interface.…”

“…Here, we propose instead an alternative choice, the so-called T-coercivity theory [3,1] to solve the problem, which relies on the use of explicit inf-sup operators. Interestingly, it can also be used to prove the convergence of finite element discretizations [3,20].…”

An optimization-based numerical method for diffusion problems with sign-changing coefficients

“…On general meshes, two main approaches have been investigated by Chesnel and Ciarlet Jr. The first one consists in building a (mesh-dependent) operator T h , such that the bilinear form a is T h -coercive on V 0 pT h q (see, e.g., [13]). This kind of approach is limited by the fact that, in general, well-posedness cannot be proved for the whole range of admissible coefficients.…”

T-Coercivity for the Maxwell Problem with Sign-Changing Coefficients

“…Those scalar problems have been thoroughly investigated [5,35,7,24,2,12,3,10,13] and sharp results have been recently obtained using the simple variational technique of the T-coercivity. The problems are proved to be of Fredholm type in the classical functional framework if the contrasts (ratios of the values of σ across the interface) are outside some interval, which always contains the value −1.…”