The numerical treatment of surface intrinsic elliptic PDE presents several interesting chal-lenges over those posed on flat space. The implicit closest point method, (iCPM) is an embedding method well suited to these problems, and allows the treatment of general sur-faces, S. An extension operator brings surface bound information into the embedding space, to be constant in the surface normal direction, and allows the solution of the problem by standard methods. The positive Helmholtz equation, (c − △S ) u = f, is considered with ten-sor product barycentric Lagrange interpolation defining the extension operator and stan-dard second-order centered di˙erences for the ambient Laplacian. Under this scheme, a non-symmetric system with poor sparsity is obtained, which reduces the performance of iterative solvers and motivates the development of specialized solvers. Optimized restricted additive Schwarz (ORAS) methods are well suited to this task and are formulated for these problems. The interesting geometry of the problem presents challenges for the construction of subdomains as well as the enforcement of Robin boundary conditions. The developed solvers perform well over a range of test problems with the optimized methods providing a distinct advantage. With Krylov acceleration, convergence is obtained rapidly with dimin-ished di˙erence between the optimized and non-optimized methods.
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Thesis advisor: Ruuth, Steven
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