Stencil scaling for vector-valued PDEs with applications to generalized Newtonian fluids
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Matrix-free finite element implementations for large applications provide an attractive alternative to standard sparse matrix data formats due to the significantly reduced memory consumption. Here, we show that they are also competitive with respect to the run time if combined with suitable stencil scaling techniques. We focus on variable coefficient vector-valued partial differential equations as they arise in many physical applications. The presented method is based on scaling constant reference stencils instead of evaluating the bilinear forms on-the-fly. We provide theoretical and experimental performance estimates showing the advantages of this new approach compared to the traditional on-the-fly integration and stored matrix approaches. In our numerical experiments, we consider two specific mathematical models. Namely, linear elastostatics and incompressible Stokes flow. The final example considers a non-linear shear-thinning generalized Newtonian fluid. For this type of non-linearity, we present an efficient approach to compute a regularized strain rate which is then used to define the node-wise viscosity. In the best scenario, we could observe a speedup of about 122 on-the-fly integration. The largest considered example involved solving a Stokes problem with 12288 compute cores.
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