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A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions. (English) Zbl 1188.65153
The authors consider a two-dimensional boundary-value reaction-diffusion problem on the unit square. The boundary conditions are of Dirichlet type. The equation is then discretized with a Galerkin method using piecewise bilinear trial functions on a Shishkin mesh. The efficiency of the method is improved using a two-scale sparse grid method. It is shown that this technique achieves the same order of accuracy in the energy norm as the standard finite element scheme on the Shishkin mesh while the number of degrees of freedom can be reduced from $$O(N^2)$$ to $$O(N^{3/2})$$. Proofs of convergence for both methods are also given. Furthermore a generalization to systems of reaction-diffusion equations is presented. Numerical results for a single equation and a coupled system illustrate the efficiency of the two-scale sparse grid method.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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