Magnetic-Electric Formulations for Stationary Magnetohydrodynamics
  Models

There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

Abstract

We discuss magnetic-electric fields based finite element schemes for stationary magnetohydrodynamics (MHD) systems with two types of boundary conditions. The schemes are unconditional well-posed and stable. Moreover, magnetic Gauss's law \(\nabla\cdot\bm{B}=0\) is preserved precisely on the discrete level. We establish a key \(L^{3}\) estimate for divergence-free finite element functions for a new type of boundary condition. With this estimate and a similar one in \cite{hu2015structure}, we rigorously prove the convergence of Picard iterations and the finite element schemes. These results show that the proposed finite element methods converge for singular solutions.

Related collections

Most cited references 12

Record: found
Abstract: not found
Article: not found

Finite element exterior calculus, homological techniques, and applications

Ragnar Winther, Richard S. Falk, Douglas N. Arnold (2006)

0 comments Cited 108 times – based on 0 reviews      Review now

Bookmark

Record: found
Abstract: found
Article: found

Is Open Access

Finite element exterior calculus: from Hodge theory to numerical stability

Douglas N. Arnold, Richard S. Falk, Ragnar Winther (2009)

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.