A twoscale approach for efficient onthefly operator assembly in massively parallel high performance multigrid codes
 Abstract
Large scale matrixfree finite element implementations save memory and are often significantly faster than implementations using classical sparse matrix techniques. They are especially well suited for massively parallel geometric multigrid solvers in combination with hierarchical hybrid grids on polyhedral domains. In the case of constant coefficients, the number of different stencil entries depends only on the coarse grid and does not increase with the number of refinement levels. However, for nonpolyhedral domains the situation changes. Then even for the Laplace operator, the element mapping leads to fine grid stencils that can vary from grid point to grid point. Traditional matrixfree techniques that are based on an elementwise assembly then result in a considerably increase in computational cost. To compensate for this shortcoming, we introduce a new twoscale approach that uses a surrogate operator. It exploits a piecewise polynomial approximation of the entries of the stencil of the fine grid operator with respect to the coarse mesh size. The lowcost evaluation of these surrogate polynomials results in an efficient stencil assembly onthefly for nonpolyhedral domains. We discuss and illustrate numerically twoscale a priori bounds. The accuracy of the approximate solution can be further improved if combined with a double discretization technique. A careful performance analysis in combination with a hardware–aware code optimization based on the Execution–Cache–Memory model yields a significant speed up. Weak and strong scaling results illustrate the potential of this new twoscale approach within large scale PDE simulations.
 BibTeX

@article{id2269, author = {Bauer, S. and Mohr, M. and R\"ude, U. and Weism\"uller, J. and Wittmann, M. and Wohlmuth, B.}, doi = {10.1016/j.apnum.2017.07.006}, journal = {Applied Numerical Mathematics}, language = {en}, pages = {1438}, title = {A twoscale approach for efficient onthefly operator assembly in massively parallel high performance multigrid codes}, volume = {122}, year = {2017}, }
 EndNote

%O Journal Article %A Bauer, S. %A Mohr, M. %A Rüde, U. %A Weismüller, J. %A Wittmann, M. %A Wohlmuth, B. %R 10.1016/j.apnum.2017.07.006 %J Applied Numerical Mathematics %G en %P 1438 %T A twoscale approach for efficient onthefly operator assembly in massively parallel high performance multigrid codes %V 122 %D 2017