Mathematical Sciences: Algorithms and Numerical Analysis forPartial Differential Equations

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This award will support the efforts of the principal investigators to continue their studies of finite element methods for the numerical solution of partial differential equations. In particular, the researchers will construct preconditioners that will enable them to solve the resulting systems of algebraic equations more efficiently. The theoretical insights gained from their studies should lead to the development of improved software for use on parallel computers, which in turn will lead to improved solutions of a host of problems in two and three space dimensions. The computer has changed forever the way we go about finding solutions of partial differential equations that arise in science and technology. Unfortunately the ability of even the most powerful supercomputer to solve complicated systems is limited very often by the sheer size of the algebraic equations that result from discretizing the differential equations. In order to overcome this "size" limitation numerical analysts have devised ways to "precondition" these algebraic equations so as to make them more digestible to the computer. The principal investigators on this grant will seek better ways to achieve this preconditioning.