Convergence analysis of an hp finite element method for singularly perturbed transmission problems in smooth domains

Abstract

We consider a two‐dimensional singularly perturbed transmission problem with two different diffusion coefficients, in a domain with smooth (analytic) boundary. The solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high and interface layers along the interface. Utilizing existing and newly derived regularity results for the exact solution, we prove the robustness of an hp finite element method for its approximation. Under the assumption of analytic input data, we show that the method converges at an “exponential” rate, provided the mesh and polynomial degree distribution are chosen appropriately. Numerical results illustrating our theoretical findings are also included. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Robust coupling of DPG and BEM for a singularly perturbed transmission problem

“…Literature on the numerical analysis of singularly perturbed transmission problems is scarce. In [25], Nicaise and Xenophontos study finite elements for a one-dimensional transmission problem with different diffusion coefficients, and in [26] they consider the hp-version of the FEM for a transmission problem in two dimensions. Here, the key point is an asymptotic expansion of the solution, based on the assumption of smooth geometry and analytic data.…”

An $hp$-FEM for Singularly Perturbed Transmission Problems

“…The curve Σ denotes the interface between the two subdomains and consists of the perimeter of the circle with radius 2. As in the one-dimensional case, the solution will exhibit a boundary layer along ∂Ω + and an interface layer along Σ (inside Ω + ) [14].…”

“…We consider the following singularly perturbed transmission problem from [14]: Let Ω + and Ω − be smooth domains in R 2 , with respective boundaries ∂Ω + and ∂Ω − , such that ∂Ω + ∩∂Ω − = Σ. We assume that ∂Ω is an analytic curve, i.e.…”