arXiv subjects

Michael Schomburg

Publications and source records attributed to Michael Schomburg.

6 recordsLinked to original sources

arXiv2026

New a posteriori error estimates for full-space transmission problems

In the present work, we derive functional upper bounds for the potential error arising from finite-element boundary-element coupling formulations for a nonlinear Poisson-type transmission problem. The proposed a posteriori error estimates are independent of the precise discretization scheme and provide guaranteed upper bounds for the potential error. The computation of these upper bounds is based on the solutions of local auxiliary finite element problems on patches in the interior domain and in a strip domain along the coupling boundary. Numerical experiments illustrate the performance of the proposed error estimation strategy for a related adaptive mesh-refinement strategy.

math.NA↗

arXiv2023

Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes

We show that the biharmonic Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. This paper extends recent results of the authors on the de Rham and elasticity Hilbert complexes with mixed boundary conditions and results of Pauly and Zulehner on the biharmonic Hilbert complex with empty or full boundary conditions.

math.AP↗

arXiv2021

Hilbert Complexes with Mixed Boundary Conditions -- Part 1: De Rham Complex

We show that the de Rham Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are proved as well.

math.AP↗

arXiv2021

Hilbert Complexes with Mixed Boundary Conditions -- Part 2: Elasticity Complex

We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are proved as well. This paper extends recent results on the de Rham Hilbert complex with mixed boundary conditions from [11] and recent results on the elasticity Hilbert complex with empty or full boundary conditions from [15].

math.AP↗

arXiv2019

Weck's Selection Theorem: The Maxwell Compactness Property for Bounded Weak Lipschitz Domains with Mixed Boundary Conditions in Arbitrary Dimensions

It is proved that the space of differential forms with weak exterior and co-derivative, is compactly embedded into the space of square integrable differential forms. Mixed boundary conditions on weak Lipschitz domains are considered. Furthermore, canonical applications such as Maxwell estimates, Helmholtz decompositions and a static solution theory are proved. As a side product and crucial tool for our proofs we show the existence of regular potentials and regular decompositions as well.

math.AP↗

arXiv2019

The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

For a bounded weak Lipschitz domain we show the so called `Maxwell compactness property', that is, the space of square integrable vector fields having square integrable weak rotation and divergence and satisfying mixed tangential and normal boundary conditions is compactly embedded into the space of square integrable vector fields. We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exterior domains formulated in weighted Sobolev spaces are straight forward.

math.AP↗