Parallel-vector computation with high-\(p\) element-by-element methods. (English) Zbl 0760.65097

The authors consider finite element approximations of a class of linear elliptic problems. An element-by-element (EBE) approach to the finite element system is arrived at by writing the equations explicitly in terms of element stiffness matrices and connectivity matrices. Thus it becomes apparent that the element contributions can be computed independently in parallel, and that vectorization within elements can be exploited. This approach is investigated, with a preconditioned bi-conjugate gradient method used to solve the EBE scheme. The authors make use of the \(p\)- version of the finite element method, in which the mesh is kept fixed while the degree of the polynomial basis functions is increased to obtain convergence. Some attention is given to the difficulties which have to be overcome in vectorizing the matrix-vector products, which are evaluated by breaking them up into steps. Each step is treated appropriately, by a combination of vectorizing within each element and either parallelizing over elements, or treating elements sequentially. Results of numerical studies are presented for the Helmholtz equation, and the stream function-vorticity formulation of the 2-D Navier-Stokes equations. Full details of numerical experiments are given, together with an assessment of the performance of the schemes used. The results indicate a high degree of parallelization and vectorization, which are introduced in a natural way, for high \(p\) (\(p=6\), say).


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
65F10 Iterative numerical methods for linear systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q30 Navier-Stokes equations
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