Hannover, Univ., Diss., 2004.

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.

We present an $hp$-adaptive strategy that is based on estimating the decay of the expansion coefficients when a function is expanded in $L^2$-orthogonal polynomails on a triangle or a tetrahedron. This method is justified by showing that the decay of the coefficients is exponential if and only if the function is analytic. Numerical examples illustrate the performance of this approach, and we compare it with two other $hp$-adaptive strategies.

We analyze and compare different techniques to set up the stiffness matrix in the hp-version of the finite element method. The emphasis is on methods for second order elliptic problems posed on meshes including triangular and tetrahedral elements. The polynomial degree may be variable. We present a generalization of the Spectral Galerkin Algorithm of [7], where the shape functions are adapted to the quadrature formula, to the case of triangles/tetrahedra. Additionally, we study on-the-fly matrix-vector multiplications, where merely the matrix-vector multiplication is realized without setting up the stiffness matrix. Numerical studies are included.

The boundary concentrated finite element method is a variant of the hp-version of the finite element method that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and low regularity boundary conditions. For this method we present two multilevel preconditioners that lead to preconditioned stiffness matrices with condition numbers that are bounded uniformly in the problem size N. The cost of applying the preconditioners is O(N). Numerical examples illustrate the efficiency of the algorithms.

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.

We present an $hp$-adaptive strategy that is based on estimating the decay of the expansion coefficients when a function is expanded in $L^2$-orthogonal polynomails on a triangle or a tetrahedron. This method is justified by showing that the decay of the coefficients is exponential if and only if the function is analytic. Numerical examples illustrate the performance of this approach, and we compare it with two other $hp$-adaptive strategies.

We analyze and compare different techniques to set up the stiffness matrix in the hp-version of the finite element method. The emphasis is on methods for second order elliptic problems posed on meshes including triangular and tetrahedral elements. The polynomial degree may be variable. We present a generalization of the Spectral Galerkin Algorithm of [7], where the shape functions are adapted to the quadrature formula, to the case of triangles/tetrahedra. Additionally, we study on-the-fly matrix-vector multiplications, where merely the matrix-vector multiplication is realized without setting up the stiffness matrix. Numerical studies are included.

The boundary concentrated finite element method is a variant of the hp-version of the finite element method that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and low regularity boundary conditions. For this method we present two multilevel preconditioners that lead to preconditioned stiffness matrices with condition numbers that are bounded uniformly in the problem size N. The cost of applying the preconditioners is O(N). Numerical examples illustrate the efficiency of the algorithms.