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

04/05/2019
by   Dmytro Shulga, et al.
0

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).

READ FULL TEXT
research
09/20/2021

A Multivariate Spline based Collocation Method for Numerical Solution of Partial Differential Equations

We propose a collocation method based on multivariate polynomial splines...
research
02/23/2021

Solving high-dimensional parabolic PDEs using the tensor train format

High-dimensional partial differential equations (PDEs) are ubiquitous in...
research
02/28/2021

A recursive system-free single-step temporal discretization method for finite difference methods

Single-stage or single-step high-order temporal discretizations of parti...
research
06/05/2023

Deep Generalized Green's Functions

In this study, we address the challenge of obtaining a Green's function ...
research
02/21/2023

Data-based Adaptive Refinement of Finite Element Thin Plate Spline

The thin plate spline, as introduced by Duchon, interpolates a smooth su...
research
09/03/2020

A two level method for isogeometric discretizations

Isogeometric Analysis (IGA) is a computational technique for the numeric...

Please sign up or login with your details

Forgot password? Click here to reset