Sparse grid methods for singularly perturbed problems
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This thesis is concerned with the design, analysis and implementation of sparse grid finite element methods applied to singularly perturbed partial differential equations, in two and three dimensions. Typically, sparse grid methods are constructed using a hierarchical grid approach. This thesis presents a two-dimensional multiscale sparse grid method that is the same, up to choice of basis, as standard hierarchical sparse grid methods. However, since the method is described as a generalisation of the two-scale sparse grid method, both the analysis and implementation are significantly simplified. We provide an analysis for a multiscale sparse grid method applied to an elliptic partial differential equation, by first providing a concise expression for the difference between two multiscale interpolation operators at successive levels and then deriving a bound on this expression. The solutions to the singularly perturbed problems that we study possess boundary layers. The most commonly used numerical methods for computing solutions that resolve these layers involve layer adapted meshes, and the mesh of Shishkin in particular. We show how to apply sparse grid methods to both reaction-diffusion, and convection-diffusion problems with exponential layers, both in two dimensions. The multiscale analysis we have developed allows us to prove robust convergence. We then extend the methods to three dimensions. We provide the first (that we know of) complete numerical analysis of a standard Galerkin finite element method applied to a singularly perturbed reaction-diffusion problem in three dimensions. Moreover, we provide the first analysis for any sparse grid method applied to a singularly perturbed problem in three dimensions. We describe a two-scale sparse grid method in three dimensions, and provide a full numerical analysis for it applied to a singularly perturbed reaction-diffusion problem. This requires a suitable three-dimensional Shishkin mesh, solution decomposition and bounds on derivatives of its components, which are all presented in detail. As with the two-scale sparse grid method in two dimensions, an expression for the difference between the standard trilinear interpolation operator and the two-scale interpolation operator is the key to completing the analysis.
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