Real and Complex Analysis

Finite element exterior calculus is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are re- vealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Lapla- cian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

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Finite element exterior calculus, homological techniques, and applications

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

https://perso.univ-rennes1.fr/martin.costabel/publis/BuCo_Trace_JMAA.pdf

On traces for H(curl,Ω) in Lipschitz domains

Weakly Differentiable Functions

Let $X$ be a metric space with doubling measure, and $L$ be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on $L^2(X)$. In this article the authors present a theory of Hardy and BMO spaces associated to $L$, including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that $L$ is a Schrodinger operator on $\mathbb{R}^n$ with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces $H^p_L(X)$ for $p>1$, which may or may not coincide with the space $L^p(X)$, and show that they interpolate with $H^1_L(X)$ spaces by the complex method.

https://faculty.missouri.edu/~hofmanns/papers/HLMMY_final.pdf

Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

Introduction Singular integrals on Lipschitz submanifolds of codimension one Estimates on fundamental solutions General second-order strongly elliptic systems The Dirichlet problem for the Hodge Laplacian and related operators Natural boundary problems for the Hodge Laplacian in Lipschitz domains Layer potential operators on Lipschitz domains Rellich type estimates for differential forms Fredholm properties of boundary integral operators on regular spaces Weak extensions of boundary derivative operators Localization arguments and the end of the proof of Theorem 6.2 Harmonic fields on Lipschitz domains The proofs of the Theorems 5.1-5.5 The proofs of the auxiliary lemmas Applications to Maxwell's equations on Lipschitz domains Analysis on Lipschitz manifolds The connection between $d_\partial$ and $d_{\partial\Omega}$ Bibliography.

Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for Lp-integrable bounday data for optimal values of p’s. We also discuss a number of relevant applications in electromagnetic scattering.

Vector potential theory on nonsmooth domains in R3 and applications to electromagnetic scattering

We formulate and solve the Poisson problem for the exterior
derivative operator with Dirichlet boundary condition in
Lipschitz domains, of arbitrary topology, for data in
Besov and Triebel-Lizorkin spaces.

https://hal.archives-ouvertes.fr/hal-00535692/document

Dorina Mitrea

Papers

Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds

Vector potential theory on nonsmooth domains in R3 and applications to electromagnetic scattering

The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains

Extending Sobolev functions with partially vanishing traces from locally (ε,δ)-domains and applications to mixed boundary problems