Finite element methods for Maxwell's equations.

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict nonpolar size effects. It is intended for the homogenized description of highly heterogeneous, but nonpolar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models, our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models.

/pdf/a-unifying-perspective-the-relaxed-linear-micromorphic-s3dvc7yf5l.pdf

A unifying perspective: the relaxed linear micromorphic continuum

Dynamic problems for metamaterials: Review of existing models and ideas for further research

R Leis 1986 Chichester: John Wiley viii + 266 pp price £2450 ISBN 0 471 90863 0 This book is an account of some recent developments in the theory of partial differential equations for readers thoroughly familiar with functional analysis It deals with various types of problem for the wave equation, Maxwell's equations, Schrodinger's equation, the plate equation etc, but dissipative systems are hardly discussed

https://iopscience.iop.org/article/10.1088/0031-9112/37/10/027/pdf

Initial Boundary Value Problems in Mathematical Physics

https://www.uni-due.de/~hm0014/ag_neff/publikation_neu/Poincare%20meets%20Korn%20via%20Maxwell-%20Extending%20Korn's%20First%20Inequality%20to%20Incompatible%20Tensor%20Fields%20201405.pdf

Poincaré meets Korn via Maxwell: Extending Korn's first inequality to incompatible tensor fields

Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative

A canonical extension of Kornʼs first inequality to H(Curl) motivated by gradient plasticity with plastic spin

Let $\Omega\subset\mathbb{R}^3$ be a bounded weak Lipschitz domain with boundary $\Gamma:=\operatorname{\partial}\Omega$ divided into two weak Lipschitz submanifolds $\Gamma_{\tau}$ and $\Gamma_{\nu}$, and let $\varepsilon$ denote an $\mathsf{L}^{\infty}$-matrix field inducing an inner product in $\mathsf{L}^2(\Omega)$. The key result of this paper is the so-called Maxwell compactness property, i.e., the Hilbert space $\bigl\{E\in\mathsf{L}^2(\Omega):\operatorname{rot}E\in\mathsf{L}^2(\Omega),\,\operatorname{div}\varepsilon E\in\mathsf{L}^2(\Omega),\,\nu\times E|_{\Gamma_{\tau}}=0,\,\nu\cdot\varepsilon E|_{\Gamma_{\nu}}=0\bigr\}$ is compactly embedded into $\mathsf{L}^2(\Omega)$. 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 straightfo...

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

For a bounded domain with connected Lipschitz boundary, we prove the existence of some c > 0, such that 






holds for all square-integrable tensor fields , having square-integrable generalized “rotation” tensor fields and vanishing tangential trace on ∂Ω, where both operations are to be understood row-wise. Here, in each row, the operator curl is the vector analytical reincarnation of the exterior derivative d in . For compatible tensor fields T, that is, T = ∇ v, the latter estimate reduces to a non-standard variant of Korn's first inequality in , namely 






for all vector fields , for which ∇ vn,n = 1, … ,N, are normal at ∂Ω. Copyright © 2012 John Wiley & Sons, Ltd.

Dirk Pauly

Papers

Poincaré meets Korn via Maxwell: Extending Korn's first inequality to incompatible tensor fields

Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative

A canonical extension of Kornʼs first inequality to H(Curl) motivated by gradient plasticity with plastic spin

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

Maxwell meets Korn: A new coercive inequality for tensor fields in RN×N with square-integrable exterior derivative