Convergence Results for Non-Conforming hp Methods: The Mortar Finite Element Method

In this paper, we present uniform convergence results for the mortar finite element method (which is an example of a non-conforming method), for h, p and hp discretizations over general meshes. Our numerical and theoretical results show that the mortar finite element method is a good candidate for hp implementation and also that the optimal rates afforded by the conforming h, p and hp discretizations are preserved when this non-conforming method is used, even over highly nonquasiuniform meshes. Design over complex domains often requires the concatenation of separately constructed meshes over subdomains. In such cases it is difficult to coordinate the submeshes so that they conform over interfaces. Therefore, non-conforming elements such as the mortar finite element method [2, 3, 4] are used to “glue” these submeshes together. Such techniques are also useful in applications where the discretization needs to be selectively increased in localized regions (such as those around corners or other features) which contribute most to the pollution error in any problem. Moreover, different variational problems in different subdomains can also be combined using non-conforming methods. When p and hp methods are being used, the interface incompatibility may be present not only in the meshes but also in the degrees chosen on the elements from the two sides. Hence the concatenating method used must be formulated to accomodate various degrees, and also be stable and optimal both in terms of mesh refinement (h version) and degree enhancement (p version). Moreover, this stability and optimality should be preserved when highly non-quasiuniform meshes are used around corners (such as the geometrical ones in the hp version). We present theoretical convergence results for the mortar finite element method from [7],[8] and extend these in two ways in this paper. First, we show that the stability estimates established for the mortar projection operator (Theorem 2 in