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A new non-conforming Petrov-Galerkin finite-element method with triangular elements for a singularly perturbed advection-diffusion problem. (English) Zbl 0806.65111
The authors present an exponentially fitted Petrov-Galerkin finite element method with triangular elements for a stationary advection- diffusion problem. The method is based on a discrete bilinear formulation and on Delaunay triangulation and its Dirichlet tessellation. An error estimate in an \(\varepsilon\)-independent discrete energy norm is included for an arbitrary Delaunay triangulation. The approximate solution is stable with respect to the discrete energy norm. The condition that the angles in the triangulation are acute is relaxed.
The coefficient matrix in this method coincides with that obtained from the exponentially fitted box method. The effectiveness of the method is shown by solving the singularly perturbed advection-diffusion problem. Even though it takes more computer time, the improvement over the upwind and central-difference method is significant.

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
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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