2017/09/20 by S. H. Bauer, Daniel Drzisga, Bauer, Simon +9
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Engineering #Electromagnetic Simulation and Numerical Methods #FOS: Computer and information sciences #FOS: Mathematics #Finance #Numerical Analysis (math.NA) #Numerical methods for differential equations #and Science (cs.CE)
paper · pdf · doi:10.48550/arxiv.1709.06793
openalex publication_date 2017/09/20 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We present a novel approach to fast on-the-fly low order finite element\nassembly for scalar elliptic partial differential equations of Darcy type with\nvariable coefficients optimized for matrix-free implementations. Our approach\nintroduces a new operator that is obtained by appropriately scaling the\nreference stiffness matrix from the constant coefficient case. Assuming\nsufficient regularity, an a priori analysis shows that solutions obtained by\nthis approach are unique and have asymptotically optimal order convergence in\nthe H1- and the L2-norm on hierarchical hybrid grids. For the\npre-asymptotic regime, we present a local modification that guarantees uniform\nellipticity of the operator. Cost considerations show that our novel approach\nrequires roughly one third of the floating-point operations compared to a\nclassical finite element assembly scheme employing nodal integration. Our\ntheoretical considerations are illustrated by numerical tests that confirm the\nexpectations with respect to accuracy and run-time. A large scale application\nwith more than a hundred billion (1.6\⋅1011) degrees of freedom\nexecuted on 14,310 compute cores demonstrates the efficiency of the new scaling\napproach.\n