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A new approximation technique for div-curl systems

Authors: James H. Bramble and Joseph E. Pasciak
Journal: Math. Comp. 73 (2004), 1739-1762
MSC (2000): Primary 65F10, 65N55
Published electronically: August 26, 2003
MathSciNet review: 2059734
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Abstract: In this paper, we describe an approximation technique for div-curl systems based in $(L^2(\Omega )^3)$ where $\Omega$ is a domain in $\mathbb {R}^3$. We formulate this problem as a general variational problem with different test and trial spaces. The analysis requires the verification of an appropriate inf-sup condition. This results in a very weak formulation where the solution space is $(L^2(\Omega ))^3$ and the data reside in various negative norm spaces. Subsequently, we consider finite element approximations based on this weak formulation. The main approach of this paper involves the development of “stable pairs" of discrete test and trial spaces. With this approach, we enlarge the test space so that the discrete inf-sup condition holds and we use a negative-norm least-squares formulation to reduce to a uniquely solvable linear system. This leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low Sobolev regularity (e.g., $(H^s(\Omega ))^3$ with $0< s< 1/2$). The resulting algebraic equations are symmetric, positive definite and well conditioned. A second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned. Some numerical results are also presented.

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Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Div-curl systems, inf-sup condition, finite element approximation, Petrov-Galerkin, negative-norm least-squares, Maxwell’s equations
Received by editor(s): January 8, 2003
Received by editor(s) in revised form: March 18, 2003
Published electronically: August 26, 2003
Additional Notes: This work was supported in part by the National Science Foundation through grants DMS-9805590 and DDS-9973328.
Article copyright: © Copyright 2003 American Mathematical Society