The boundary concentrated finite element method is a variant of the hp-version of the FEM that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and boundary conditions with low regularity or non-smooth geometries. In this paper we consider the case of the discretization of a Dirichlet problem with exact solution $u \in H^{1+\delta}(\Omega)$ and investigate the local error in various norms. We show that for a $\beta > 0$ these norms behave as $O(N^{−\delta−\beta})$, where $N$ denotes the dimension of the underlying finite element space. Furthermore, we present a new Gauss-Lobatto based interpolation operator that is adapted to the case non-uniform polynomial degree distributions.

This paper is devoted to the fast solution of interface concentrated finite element equations. The interface concentrated finite element schemes are constructed on the basis of a non-overlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain. Similar to data-sparse boundary element domain decomposition methods the total number of unknowns per subdomain behaves like $O((H/h)^{d−1})$, where H, h, and d denote the usual scaling parameter of the subdomains, the average discretization parameter of the subdomain boundaries, and the spatial dimension, respectively. We propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns. In particular, the so-called All-Floating Finite Element Tearing and Interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps.

The boundary concentrated finite element method is a variant of the hp-version of the FEM that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and boundary conditions with low regularity or non-smooth geometries. In this paper we consider the case of the discretization of a Dirichlet problem with exact solution $u \in H^{1+\delta}(\Omega)$ and investigate the local error in various norms. We show that for a $\beta > 0$ these norms behave as $O(N^{−\delta−\beta})$, where $N$ denotes the dimension of the underlying finite element space. Furthermore, we present a new Gauss-Lobatto based interpolation operator that is adapted to the case non-uniform polynomial degree distributions.

This paper is devoted to the fast solution of interface concentrated finite element equations. The interface concentrated finite element schemes are constructed on the basis of a non-overlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain. Similar to data-sparse boundary element domain decomposition methods the total number of unknowns per subdomain behaves like $O((H/h)^{d−1})$, where H, h, and d denote the usual scaling parameter of the subdomains, the average discretization parameter of the subdomain boundaries, and the spatial dimension, respectively. We propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns. In particular, the so-called All-Floating Finite Element Tearing and Interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps.