Abstract. The paper presents composition graph (CP-graph) grammar, which consists of a set of CP-graph transformations, suitable for modeling all aspects of parallel hp adaptive Finite Element Method (FEM) computations. The parallel hp adaptive FEM allows to utilize distributed computational meshes, with finite elements of various size (thus h stands for element diameter) and polynomial orders of approximation varying locally, on finite elements edges and interiors (thus p stands for polynomial order of approximation). The computational mesh is represented by attributed CP-graph. The proposed graph transformations model the initial mesh generation, procedure of h refinement (breaking selected finite elements into son elements), and p refinement (adjusting polynomial orders of approximation on selected element edges and interiors), as well as partitioning of computational mesh into sub-domains and enforcement of mesh regularity rules over the distributed data structure.

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We construct quasi-optimal elimination trees for 2D finite element meshes with singularities. These trees minimize the complexity of the solution of the discrete system. The computational cost estimates of the elimination process model the execution of the multifrontal algorithms in serial and in parallel shared-memory executions. Since the meshes considered are a subspace of all possible mesh partitions, we call these minimizers quasi-optimal. We minimize the cost functionals using dynamic programming. Finding these minimizers is more computationally expensive than solving the original algebraic system. Nevertheless, from the insights provided by the analysis of the dynamic programming minima, we propose a heuristic construction of the elimination trees that has costONelog⁡Ne, whereNeis the number of elements in the mesh. We show that this heuristic ordering has similar computational cost to the quasi-optimal elimination trees found with dynamic programming and outperforms state-of-the-art alternatives in our numerical experiments.

Abstract. The paper presents composition graph (CP-graph) grammar, which consists of a set of CP-graph transformations, suitable for modeling triangular finite element mesh transformations utilized by the self-adaptive hp Finite Element Method (FEM). The hp adaptive FEM allows to utilize distributed computational meshes, with finite elements of various size (thus h stands for element diameter) and polynomial orders of approximation varying locally, on finite elements edges and interiors (thus p stands for polynomial order of approximation). The computational triangular mesh is represented by attributed CP-graph. The proposed graph transformations model the initial mesh generation, procedure of h refinement (breaking selected finite elements into son elements), and p refinement (adjusting polynomial orders of approximation on selected element edges and interiors). The graph grammar has been defined and verified by implemented graph grammar transformation software tool.

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