Volume 17, Issue 4

We consider a one-dimensional singularly perturbed transmission problem with two different diffusion coefficients. The solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high. We derive and analyze various finite element approaches for the approximation of the solution and conduct numerical computations that show the robustness (or lack thereof) of each approach.

The all-floating Boundary Element Tearing and Interconnecting method incooperates the Dirichlet boundary conditions by additional constraints in the dual formulation of the standard Tearing and Interconnecting methods. This simplifies the implementation, as all subdomains are considered as floating subdomains. The method shows an improved asymptotic complexity compared to the standard BETI approach. The all-floating BETI method is presented for linear elasticity in this paper.

This paper presents a posteriori finite element error estimates for Signorini's problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The a posteriori error control relies on estimating the discretization error of an auxiliary problem which is given as a variational equation. The estimation consists of error bounds for the discretization error of the auxiliary problem and some further terms which capture the geometrical error and the error in the complementary condition. The derived estimates are applied to h - and hp -adaptive refinement and enrichment strategies. Numerical results confirm the applicability of the theoretical findings. In particular, optimal algebraic and almost exponential convergence rates are obtained.

An implicit difference scheme is presented for a space–time fractional convection–diffusion equation. The equation is obtained from the classical integer order convection–diffusion equations with fractional order derivatives for both space and time. First-order consistency, unconditional stability, and first-order convergence of the method are proven using a novel shifted version of the classical Grünwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.