Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther \cite{ArFaWi2010} includes a well-developed theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for Arnold-Falk-Winther mixed finite element methods for Hodge-de Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell's equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian.
Conforming and Non-Conforming Functional A Posteriori Error Estimates for Elliptic Boundary Value Problems in Exterior Domains: Theory and Numerical Tests
The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in a previous paper by the authors for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh of the domain, we build a map which decomposes piecewise smooth $k$-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are t…
The auxiliary space preconditioner for the de Rham complex
Portland State University（波特兰州立大学）；Institute of Computational Mathematics, Johannes Kepler University（约翰内斯·开普勒大学计算数学研究所）；Center for Applied Scientific Computing, LLNL（劳伦斯利弗莫尔国家实验室应用科学计算中心）
This paper proposes a finite element method for solving the periodic steady-state problem for the scalar-valued and vector-valued Poisson equations, a simple reduction model of the Maxwell equations under the Coulomb gauge. Introducing a new potential variable, we reformulate two systems composed of the scalar-valued and vector-valued Poisson problems to a single Hodge-Laplace problem for the 1-form in $\mathbb{R}^4$ using the standard de Rham complex. Consequently, we can directly apply the Finite Element Exterior Calculus (FEEC) theory in $\mathbb{R}^4$ to deduce the well-posedness, stabilit…
In this paper, we present several new a posteriori error estimators and two adaptive mixed finite element methods \textsf{AMFEM1} and \textsf{AMFEM2} for the Hodge Laplacian problem in finite element exterior calculus. We prove that \textsf{AMFEM1} and \textsf{AMFEM2} are both convergent starting from any initial coarse mesh. A suitably defined quasi error is crucial to the convergence analysis. In addition, we prove the optimality of \textsf{AMFEM2}. The main technical contribution is a localized discrete upper bound. As opposed to existing literature, our results work on Lipschitz domains wi…
This paper is concerned with a numerical solution to the scattering of a time-harmonic electromagnetic wave by a bounded and impenetrable obstacle in three dimensions. The electromagnetic wave propagation is modeled by a boundary value problem of Maxwell's equations in the exterior domain of the obstacle. Based on the Dirichlet-to-Neumann (DtN) operator, which is defined by an infinite series, an exact transparent boundary condition is introduced and the scattering problem is reduced equivalently into a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is …
Energy-preserving numerical methods for solving the Hodge wave equation is developed in this paper. Based on the de Rham complex, the Hodge wave equation can be formulated as a first-order system and mixed finite element methods using finite element exterior calculus is used to discretize the space. A continuous time Galerkin method, which can be viewed as a modification of the Crank-Nicolson method, is used to discretize the time which results in a full discrete method preserving the energy exactly when the source term is vanished. A projection based operator is used to establish the optimal …
In mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex, the exterior derivative operators are computed exactly, so the spatial locality is preserved. However, the numerical approximations of the associated coderivatives are nonlocal and it can be regarded as an undesired effect of standard mixed methods. For numerical methods with local coderivatives a perturbation of low order mixed methods in the sense of variational crimes has been developed for simplicial and cubical meshes. In this paper we extend the low order method to all high orders on cub…
Consider the elastic scattering of an incident wave by a rigid obstacle in three dimensions, which is formulated as an exterior problem for the Navier equation. By constructing a Dirichlet-to-Neumann (DtN) operator and introducing a transparent boundary condition, the scattering problem is reduced equivalently to a boundary value problem in a bounded domain. The discrete problem with the truncated DtN operator is solved by using the a posteriori error estimate based adaptive finite element method. The estimate takes account of both the finite element approximation error and the truncation erro…
Local coderivatives and approximation of Hodge Laplace problems
The Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校计算科学与工程研究所）；Department of Mathematics, University of Oslo（奥斯陆大学数学系）
We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in [http://dx.doi.org/10.1090/mcom/3079, arXiv:2109.02553, our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of th…
Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite element methods. Multisymplectic variational integrators are a class of geometric numerical integrators for Lagrangian and Hamiltonian field theories, and they yield methods that preserve the multisymplectic structure and momentum-conservation properties of the continuous system. In this paper, we investigate the synthesis of these two approaches, by constructin…
