The local Green’s function method in singularly perturbed convection-diffusion problems. (English) Zbl 1198.65058
Summary: Previous theoretical and computational investigations have shown high efficiency of the local Green’s function method for the numerical solution of singularly perturbed problems with sharp boundary layers. However, in several space variables those functions, used as projectors in the Petrov-Galerkin scheme, cannot be derived in a closed analytical form. This is an obstacle for the application of the method when applied to multi-dimensional problems. The present work proposes a semi-analytical approach to calculate the local Green’s function, which opens a way to effective practical application of the method. Besides very accurate approximation, the matrix stencils obtained with these functions allow the use of fast and stable iterative solutions of the large sparse algebraic systems that arise from the grid-discretization. The advantages of the method are illustrated by numerical examples.
MSC:
| 65F10 | Iterative numerical methods for linear systems |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 65R10 | Numerical methods for integral transforms |
| 65R20 | Numerical methods for integral equations |
| 35K57 | Reaction-diffusion equations |
Keywords:
convection-diffusion equation; Petrov-Galerkin discretization; Fourier transform; integral equations; iterative solutionReferences:
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