Elliptic Solvers with Adaptive Mesh Refinement on Complex Geometries

Adaptive mesh refinement (AMR) computations are complicated by their dynamic nature. The development of elliptic solvers for realistic applications is complicated by both the complexity of the AMR and the geometry of realistic problem domains. For example, the inclusion of AMR within the solution process on non-trivial curvilinear coordinate grids forces the refinement process to include some grid generation issues. Invariably, such geometric information, the mapping from which the original curvilinear coordinate grid was built, is unavailable in the solution process which would otherwise not require it. For this reason, most AMR work is done exclusively on rectangular Cartesian grids.

Research on multigrid based elliptic solvers for adaptive grids (FAC, AFAC, AFACx solvers) in complex geometries will be presented. The elliptic solvers use the AMR++ library, an object-oriented library for the development of adaptive mesh refinement applications. AMR++ uses the Overture framework for the independent handling of the overlapping grid geometry. Overture is an object-oriented framework for the solution of PDEs in complex geometries. The adaptive elliptic solvers take advantage of numerous objects from the Overture framework including array objects, grid objects, and operator libraries which define high level operators on curvilinear coordinate grids (e.g. div, grad, curl, etc.). It will additionally be shown how the use of this approach makes otherwise intractable applications relatively simple to build.

In this talk we will present the use of adaptive mesh refinement for the solution of elliptic boundary value problems such as the Poisson, reaction-diffusion, and Stokes equations with Dirichlet boundary data posed as First-Order System Least Squares (FOSLS) Systems. In the solution of complex problems in complex geometries reliably estimating regions of the computational domain where refinement is required during the AMR solution process is in itself an area of research. FOSLS methodologies generate reliable local error estimators, and hence are extremely useful when incorporated into the AMR solution process. Numerical results for applying AFACx to solving systems of elliptic equations arising from FOSLS methodologies on curvilinear AMR grids will be presented. Prior to this work AFACx has only been applied to solving the scalar, constant coefficient second order elliptic diffusion equation.

Object-oriented methodologies in scientific computing provide a means for rapid development of extremely complex scientific applications. However, there is often a loss in performance associated with introducing these abstractions. This arises from the inability of the compiler to optimize user defined abstractions. Automatic transformation of user level abstractions into code that the compiler can optimize is another area of research that the author has been involved in. I shall briefly describe advances in this direction also.

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