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

TL;DR

This paper presents Tensor B-spline methods as a high-performance, efficient, and mesh-free alternative to FEM for solving PDEs, demonstrating superior computational efficiency and accuracy in large-scale problems.

Contribution

It introduces the principles of Tensor B-spline methodology, analyzes their performance, and demonstrates their advantages over FEM in large-scale PDE solutions.

Findings

Tensor B-spline methods achieve high accuracy in PDE approximation.

They significantly outperform FEM in computational efficiency and memory usage.

Successful large-scale heat-equation solution with 0.8 billion nodes on heterogeneous hardware.

Abstract

Tensor B-spline methods are a high-performance alternative to solve partial differential equations (PDEs). This paper gives an overview on the principles of Tensor B-spline methodology, shows their use and analyzes their performance in application examples, and discusses its merits. Tensors preserve the dimensional structure of a discretized PDE, which makes it possible to develop highly efficient computational solvers. B-splines provide high-quality approximations, lead to a sparse structure of the system operator represented by shift-invariant separable kernels in the domain, and are mesh-free by construction. Further, high-order bases can easily be constructed from B-splines. In order to demonstrate the advantageous numerical performance of tensor B-spline methods, we studied the solution of a large-scale heat-equation problem (consisting of roughly 0.8 billion nodes!) on a heterogeneous workstation consisting of multi-core CPU and GPUs. Our experimental results nicely confirm the excellent numerical approximation properties of tensor B-splines, and their unique combination of high computational efficiency and low memory consumption, thereby showing huge improvements over standard finite-element methods (FEM).