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Dirk Pauly

Publications and source records attributed to Dirk Pauly.

At least 37 records · Page 2Linked to original sources
arXiv2020

The Index of Some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fields

We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the help of explicitly describing the dimension of the cohomology groups of generalised (`harmonic') Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac-type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and for the theory of general relativity. The differential operators are of mixed order and cannot be seen as leading order type with relatively compact perturbation. As a by-product we present a comprehensive description of the underlying generalised Dirichlet-Neumann vector and tensor fields defining the respective cohomology groups, including an explicit construction of bases in terms of topological invariants, which are of both analytical and numerical interest. Though being defined by certain projection mechanisms, we shall present a way of computing these basis functions by solving certain PDEs given in variational form. For all of this we rephrase core arguments in the work of Rainer Picard [1982] applied to the de Rham complex and use them as a blueprint for the more involved cases presented here. In passing, we also provide new vector-analytical estimates of generalised Poincare-Friedrichs type useful for elasticity or the theory of general relativity.

math.FA↗
arXiv2020

Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments

We give some theoretical as well as computational results on Laplace and Maxwell constants. Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology. Our numerical aspects are presented by examples for the de Rham complex in 2D and 3D which not only confirm our theoretical findings but also indicate some interesting conjectures.

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

Time-Harmonic Electro-Magnetic Scattering in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions

This paper treats the time-harmonic electro-magnetic scattering or radiation problem governed by Maxwell's equations in an exterior weak Lipschitz domain divided into two disjoint weak Lipschitz parts We will present a solution theory using the framework of polynomially weighted Sobolev spaces for the rotation and divergence. We will show a Fredholm alternative type result to hold using the principle of limiting absorption introduced by Eidus in the 1960's. The necessary a-priori-estimate and polynomial decay of eigenfunctions for the Maxwell equations will be obtained by transferring well known results for the Helmholtz equation using a suitable decomposition of the electro-magnetic fields. The crucial point for existence is a local version of Weck's selection theorem, also called Maxwell compactness property.

math.AP↗
arXiv2019

A Short Note on a Weighted Friedrichs Inequality

In this short note we derive, for bounded domains, an upper bound for a Friedrichs type constant in a weighted Friedrichs type inequality. This upper bound generalizes a well known upper bound of the Friedrichs constant. This upper bound is also used to improve an upper bound of a Maxwell type constant for convex domains in $\mathbb{R}^3$. A simple numerical application is also given: we apply the main result in a posteriori error estimation for an elliptic problem.

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↗
arXiv2018

A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains

We prove a global version of the so-called div-curl-lemma, a crucial result for compensated compactness and in homogenization theory, for mixed tangential and normal boundary conditions in bounded weak Lipschitz domains in 3D and weak Lipschitz interfaces. The crucial tools and the core of our arguments are the de Rham complex and Weck's selection theorem, the essential compact embedding result for Maxwell's equations.

math.AP↗
arXiv2018

A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains and a Corresponding Generalized $\mathrm{A}_{0}^{*}$-$\mathrm{A}_{1}$-Lemma in Hilbert Spaces

We prove global and local versions of the so-called div-curl-lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.

math.AP↗
arXiv2018

On the Maxwell and Friedrichs/Poincare Constants in ND

We prove that for bounded and convex domains in arbitrary dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincare's constants, respectively. Especially, the second positive Maxwell eigenvalues in ND are bounded from below by the square root of the second Neumann-Laplace eigenvalue.

math.AP↗
arXiv2018

The Stationary Stokes Problem in Exterior Domains: Estimates of the Distance to Solenoidal Fields and Functional A Posteriori Error Estimates

This paper is concerned with the analysis of the inf-sup condition arising in the stationary Stokes problem in exterior domains. We deduce values of the constant in the stability lemma, which yields fully computable estimates of the distance to the set of divergence free fields defined in exterior domains. Using these estimates we obtain computable majorants of the difference between the exact solution of the Stokes problem in exterior domains and any approximation from the admissible (energy) class of functions satisfying the Dirichlet boundary condition exactly.

math.NA↗
arXiv2018

Solution Theory, Variational Formulations, and Functional A Posteriori Error Estimates for General First Order Systems with Applications to Electro-Magneto-Statics and More

We prove a comprehensive solution theory using tools from functional analysis, show corresponding variational formulations, and present functional a posteriori error estimates for general linear first order systems. As a prototypical application we will discuss the system of electro-magneto statics with mixed tangential and normal boundary conditions. Second order systems will be considered as well.

math.NA↗
arXiv2017

An Elementary method of Deriving A Posteriori Error Equalities and Estimates for Linear Partial Differential Equations

In this paper we present a simple method of deriving a posteriori error equalities and estimates for linear elliptic and parabolic partial differential equations. The error is measured in a combined norm taking into account both the primal and dual variables. We work only on the continuous (often called functional) level and do not suppose any specific properties of numerical methods and discretizations.

math.NA↗
arXiv2017

On Closed and Exact Grad-grad- and div-Div-Complexes, Corresponding Compact Embeddings for Tensor Rotations, and a Related Decomposition Result for Biharmonic Problems in 3D

It is shown that the first biharmonic boundary value problem on a topologically trivial domain in 3D is equivalent to three (consecutively to solve) second-order problems. This decomposition result is based on a Helmholtz-like decomposition of an involved non-standard Sobolev space of tensor fields and a proper characterization of the operator div-Div acting on this space. Similar results for biharmonic problems in 2D and their impact on the construction and analysis of finite element methods have been recently published by the second author. The discussion of the kernel of div-Div leads to (de Rham-like) closed and exact Hilbert complexes, the div-Div-complex and its adjoint the Grad-grad-complex, involving spaces of trace-free and symmetric tensor fields. For these tensor fields we show Helmholtz type decompositions and, most importantly, new compact embedding results. Almost all our results hold and are formulated for general bounded strong Lipschitz domains of arbitrary topology. There is no reasonable doubt that our results extend to strong Lipschitz domains in arbitrary dimensions.

math.AP↗
arXiv2016

A Posteriori Error Analysis for the Optimal Control of Magneto-Static Fields

This paper is concerned with the analysis and numerical analysis for the optimal control of first-order magneto-static equations. Necessary and sufficient optimality conditions are established through a rigorous Hilbert space approach. Then, on the basis of the optimality system, we prove functional a posteriori error estimators for the optimal control, the optimal state, and the adjoint state. 3D numerical results illustrating the theoretical findings are presented.

math.NA↗