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Simplifying numerical solution of constrained PDE systems through involutive completion. (English) Zbl 1078.35010
Starting from elliptic systems in the sense of Douglis and Nirenberg with boundary conditions satisfying the Shapiro-Lopatinskij condition, the authors show on examples of fluid flow equations the advantage of the involutive form of those systems for their numerical solution. (For the definition of involutivity they refer to a great amount of literature.) This leads to nonsquare (overdetermined) systems which can be made square again through the introduction of auxiliary variables. The authors argue that in this way the proper physical solution is obtained and unpleasant discretization problems like the inf-sup condition are avoided. The whole approach seems to lend itself to the development of a fully automatic solution procedure in which (especially for nonconstant coefficient or nonlinear systems) rather complicated equations may arise (too complicated for handwork) but the numerical solution can be performed by standard software.

MSC:
35A35 Theoretical approximation in context of PDEs
35G15 Boundary value problems for linear higher-order PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35N10 Overdetermined systems of PDEs with variable coefficients
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
Software:
FEMLAB
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References:
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