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Fast orthogonalization to the kernel of the discrete gradient operator with application to Stokes problem. (English) Zbl 1183.65131

Summary: We obtain a simple tensor representation of the kernel of the discrete \(d\)-dimensional gradient operator defined on tensor semi-staggered grids. We show that the dimension of the nullspace grows as \(\mathcal O(n^{d-2})\), where \(d\) is the dimension of the problem, and \(n\) is one-dimensional grid size. The tensor structure allows fast orthogonalization to the kernel. The usefulness of such procedure is demonstrated on three-dimensional Stokes problem, discretized by finite differences on semi-staggered grids, and it is shown by numerical experiments that the new method outperforms usually used stabilization approach.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35Q30 Navier-Stokes equations
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