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Jens Markus Melenk

Publications and source records attributed to Jens Markus Melenk.

60 records · Page 4Linked to original sources
arXiv2014

On the stability of the boundary trace of the polynomial L^2-projection on triangles and tetrahedra (extended version)

For the reference triangle or tetrahedron $T$, we study the stability properties of the $L^2(T)$-projection $Π_N$ onto the space of polynomials of degree $N$. We show $\|Π_N u\|_{L^2(\partial T)}^2 \leq C \|u\|_{L^2(T)} \|u\|_{H^1(T)}$. This implies optimal convergence rates for the approximation error $\|u - Π_N u\|_{L^2(\partial T)}$ for all $u \in H^k(T)$, $k > 1/2$.

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arXiv2013

H-matrix approximability of the inverses of FEM matrices

We study the question of approximability for the inverse of the FEM stiffness matrix for (scalar) second order elliptic boundary value problems by blockwise low rank matrices such as those given by the H-matrix format. We show that exponential convergence in the local block rank can be achieved. We also show that exponentially accurate LU-decompositions in the H-matrix format are possible for the stiffness matrices arising in the FEM. Unlike prior works, our analysis avoids any coupling of the block rank r and the mesh width h and also covers mixed Dirichlet-Neumann-Robin boundary conditions.

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arXiv2012

Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

We consider a (possibly) nonlinear interface problem in 2D and 3D, which is solved by use of various adaptive FEM-BEM coupling strategies, namely the Johnson-Nédélec coupling, the Bielak-MacCamy coupling, and Costabel's symmetric coupling. We provide a framework to prove that the continuous as well as the discrete Galerkin solutions of these coupling methods additionally solve an appropriate operator equation with a Lipschitz continuous and strongly monotone operator. Therefore, the coupling formulations are well-defined, and the Galerkin solutions are quasi-optimal in the sense of a Céa-type lemma. For the respective Galerkin discretizations with lowest-order polynomials, we provide reliable residual-based error estimators. Together with an estimator reduction property, we prove convergence of the adaptive FEM-BEM coupling methods. A key point for the proof of the estimator reduction are novel inverse-type estimates for the involved boundary integral operators which are advertized. Numerical experiments conclude the work and compare performance and effectivity of the three adaptive coupling procedures in the presence of generic singularities.

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arXiv2012

Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: proofs

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< ε\le μ\le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have \emph{boundary layers} which overlap and interact, based on the relative size of $ε$ and $% μ$. We construct full asymptotic expansions together with error bounds that cover the complete range $0 < ε\leq μ\leq 1$. For the present case of analytic input data, we derive derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.

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arXiv2011

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation.

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