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Bauer, S.; Drzisga, D.; Mohr, M.; Ruede, U.; Waluga, C. and Wohlmuth, B. (2018): A Stencil Scaling Approach For Accelerating Matrix-Free Finite Element Implementations. In: Siam Journal on Scientific Computing, Vol. 40, No. 6: C748-C778

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present a novel approach to fast on-the-fly low order finite element assembly for scalar elliptic partial differential equations of Darcy type with variable coefficients optimized for matrix-free implementations. Our approach introduces a new operator that is obtained by appropriately scaling the reference stiffness matrix from the constant coefficient case. Assuming sufficient regularity, an a priori analysis shows that solutions obtained by this approach are unique and have asymptotically optimal order convergence in the H-1 - and the L-2-norms on hierarchical hybrid grids. For the preasymptotic regime, we present a local modification that guarantees uniform ellipticity of the operator. Cost considerations show that our novel approach requires roughly one-third of the floating-point operations compared to a classical finite element assembly scheme employing nodal integration. Our theoretical considerations are illustrated by numerical tests that confirm the expectations with respect to accuracy and run-time. A large scale application with more than a hundred billion (1.6 . 10(11)) degrees of freedom executed on 14 310 compute cores demonstrates the efficiency of the new scaling approach.

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