Heirarchical Basis Multigrid/ILU Algorithms for Solving Finite Element Equations

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9706090 Bank Hierarchical Basis Multigrid/ILU Algorithms for Solving Finite Element Equations Randolph E. Bank Department of Mathematics University of California at San Diego La Jolla, CA 92093 This proposal has two main components. First, we will study algebraic hierarchical basis multigrid algorithms (HBMG/ILU). These methods solve sparse sets of linear equations arising from finite difference, finite volume and finite element discretizations of partial differential equations. They are differentiated from classical HBMG and MG methods in that they do not require a coarse grid and sequence of mesh refinements. This allows application to problems with geometrically complex domains that require many elements just for the geometric definition, and problems where the adaptivity comes from moving the mesh points rather than from refinement. Preliminary numerical experiments indicate that the methods are potentially very powerful and robust. The second component of the of proposal concerns the continuing software development of the finite element program PLTMG. Various versions of this program have been in the public domain since the late 1970's, and it is widely used in education and research environments. PLTMG solves scalar, parameter dependent, nonlinear elliptic PDE's in general regions of the plane. The principle features are adaptive mesh generation, a posteriori error estimation, HBMG (soon to be HBMG/ILU) iteration for linear systems of equations, Newton's method for nonlinearities, and continuation for parameter dependencies. The code also includes an initial mesh generator, a skeleton generator, and several graphics routines. Although the name PLTMG has remained the same, typically 80% or more of the package is revised with each new release. In many systems modeled by partial differential equations, the critical phenomena occur only in a small part of the physical domain, and may move as a function of time ( e.g. as a flame front). Even with the great advances in hardware, it is not adequate to address difficult grand-challenge class problems of this type using software based on simple uniform meshes; the demands of the problem require that computing resources be focused on the regions of most interest. The motivation for adaptive mesh algorithms is that the algorithm itself can and should identify these critical regions and respond with an appropriate mesh with little or no human intervention. The ``brains'' of adaptive algorithms are a posterior error indicators, which both estimate the current error, and indicate where additional resources should be focused. The very nonuniform and unstructured meshes resulting from adaptive algorithms require sophisticated methods, such as multigrid or the proposed HBMG/ILU, to efficiently and reliably solve the resulting systems of equations. Overall, this field provides a mosaic of important and interrelated scientific questions ranging from difficult problems in mathematical analysis to difficult computational challenges in implementing these procedures on modern computer architectures.