Finite element methods for Maxwell's equations.

/pdf/numerical-approximation-methods-for-elliptic-boundary-value-4emguz66zy.pdf

Numerical Approximation Methods For Elliptic Boundary Value Problems

We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.

Finite Element Methods for Maxwell's Equations

The Projection Theorem. Theorems on Extension and Separation. Dual Spaces and Transposed Operators. The Banach Theorem and the Banach-Steinhaus Theorem. Construction of Hilbert Spaces. L ~ 2 Spaces and Convolution Operators. Sobolev Spaces of Functions of One Variable. Some Approximation Procedures in Spaces of Functions. Sobolev Spaces of Functions of Several Variables and the Fourier Transform. Introduction to Set-Valued Analysis and Convex Analysis. Elementary Spectral Theory. Hilbert-Schmidt Operators and Tensor Products. Boundary Value Problems. Differential-Operational Equations and Semigroups of Operators. Viability Kernels and Capture Basins. First-Order Partial Differential Equations. Selection of Results. Exercises. Bibliography. Index.

Applied functional analysis, by Jean-Pierre Aubin; translated by Carole Labrousse. Pp 423. £16·50. 1980. ISBN 0-471-02149-0 (Wiley)

This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in [17] for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [26]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a significant modification of the ideas from [17]. The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.

https://www.math.tamu.edu/~rdevore/publications/109.pdf

Adaptive Wavelet Methods II—Beyond the Elliptic Case

Introduction.- The Babuska-Brezzi Theory.- Raviart-Thomas Spaces.- Mixed Finite Element Methods.

A Simple Introduction to the Mixed Finite Element Method

We consider a porous medium entirely enclosed within a fluid region and present a well-posed conforming mixed finite-element method for the corresponding coupled problem. The interface conditions refer to mass conservation, balance of normal forces and the Beavers-Joseph-Saffman law, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite-element subspaces defining the discrete formulation employ Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures and continuous piecewise-linear elements for the Lagrange multiplier. We show stability, convergence and a priori error estimates for the associated Galerkin scheme. Finally, we provide several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence.

A conforming mixed finite-element method for the coupling of fluid flow with porous media flow

A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem

A Simple Introduction to the Mixed Finite Element Method: Theory and Applications

In this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Stokes and Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, and local approximation properties of the Clement interpolant and Raviart-Thomas operator. On the other hand, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

Gabriel N. Gatica

Papers

A Simple Introduction to the Mixed Finite Element Method

A conforming mixed finite-element method for the coupling of fluid flow with porous media flow

A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem

A Simple Introduction to the Mixed Finite Element Method: Theory and Applications

A Residual-Based A Posteriori Error Estimator for the Stokes-Darcy Coupled Problem