
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 nontrivial 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 objectoriented 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 objectoriented 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,
reactiondiffusion, and Stokes equations with Dirichlet boundary data
posed as FirstOrder 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.
Objectoriented 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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