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Finite element approximation of the diffusion operator on tetrahedra. (English) Zbl 0915.65111
By the authors’ abstract: Linear Galerkin finite element discretizations of the Laplace operator produce nonpositive stiffness coefficients for internal element edges of the two-dimensional Delaunay triangulations. This property, also called the positive transmissibility (PT) condition, is a prerequisite for the existence of an M-matrix and ensures that nonphysical local extrema are not present in the solution. For tetrahedral elements, it has already been shown that the linear Galerkin approach does not in general satisfy the PT condition. The authors propose a modification of the three-dimensional Galerkin scheme that, if a Delaunay triangulation is used, satisfies the PT condition for internal edges and, if further conditions on the boundary are specified, yields an M-matrix.
Reviewer: P.Burda (Praha)

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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