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Tensor B-spline methods are a high-performance alternative to solve partialdifferential equations (PDEs). This paper gives an overview on the principlesof Tensor B-spline methodology, shows their use and analyzes their performancein application examples, and discusses its merits. Tensors preserve thedimensional structure of a discretized PDE, which makes it possible to develophighly efficient computational solvers. B-splines provide high-qualityapproximations, lead to a sparse structure of the system operator representedby shift-invariant separable kernels in the domain, and are mesh-free byconstruction. Further, high-order bases can easily be constructed fromB-splines. In order to demonstrate the advantageous numerical performance oftensor B-spline methods, we studied the solution of a large-scale heat-equationproblem (consisting of roughly 0.8 billion nodes!) on a heterogeneousworkstation consisting of multi-core CPU and GPUs. Our experimental resultsnicely confirm the excellent numerical approximation properties of tensorB-splines, and their unique combination of high computational efficiency andlow memory consumption, thereby showing huge improvements over standardfinite-element methods (FEM).

Tensor B-Spline Numerical Methods for PDEs: a High-Performance Alternative to FEM