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Abstract

We design innovative fast and robust numerical solvers by exploiting tensor structure for high-order finite element discretizations of various partial differential equations (PDE). The thesis’s main scientific contribution is the careful design and implementation of cost-efficient subspace corrections concerning Schwarz smoothers on vertex patches (i.e., the union of cells sharing a common vertex). Emphasis is put on matrix-free implementations of multilevel solvers that are the method of choice in high-performance computing. Schwarz smoothers on vertex patches lead to robust numerical solvers. They theoretically enable scalability to extremely large applications on modern supercomputers. However, naïvely computing subspace corrections is prohibitively expensive, negating their mathematical benefits and those of matrix-free operators. To this end, we develop tensor product Schwarz smoothers, exploiting low-rank tensor representations of local solvers. Then, the computational effort per unknown is linear in the polynomial degree, the cost for inverting local matrices is asymptotically negligible, and their memory consumption per unknown is constant. We develop smoothing algorithms for high-order DG and H^1-conforming discretizations of the Laplacian, presenting the prototypical differential operator that preserves separability to finite element operators. We demonstrate the high computational efficiency of our implementations in terms of the number of floating-point operations, (strong) scaling behavior, and time-to-solution. The superiority of multiplicative Schwarz methods on vertex patches (MVS) over simple non-overlapping Schwarz smoothers is shown for higher-order finite elements, arising from their superior mathematical efficiency. The mathematical efficiency of restricted additive Schwarz smoothers comparable to MVS is studied, and extensions to non-Cartesian meshes are discussed. The techniques are then extended to biharmonic and Stokes problems that lack some separability assumptions for fast inversion. We still develop cost-efficient subspace approximations for multilevel C^0-IP methods. Given the stream function method, similar smoothers are applied for H^div-IP discretizations of Stokes problems in 2D: we design novel subspace corrections involving local stream functions and local pressure post-processing for Raviart-Thomas elements. Our implementations are designed with a holistic view on computational efficiency, i.e., carefully balancing arithmetic operations and data transfer to exploit the potential of modern multi-core architectures with SIMD capabilities. Thus, optimal node-level performance is achievable. The C++ software for tensor product Schwarz smoothers is publically available.