On the fast assemblage of finite element matrices with application to nonlinear heat transfer problems

TL;DR

This paper introduces a loop-free algorithm for rapidly assembling finite element matrices in nonlinear heat transfer problems, significantly reducing computational bottlenecks in nonlinear PDE solutions.

Contribution

It presents a novel, highly optimized loop-free method for finite element matrix assembly applicable to nonlinear PDEs, improving computational efficiency.

Findings

Assembly time reduced by a significant factor

Algorithm leverages optimized matrix-matrix multiplication

Applicable to nonlinear heat transfer problems

Abstract

The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is frequently pointed out as one of the bottlenecks in the computations. The second bottleneck being the large and numerous linear systems to be solved arising from the use of Newton's method to solve nonlinear systems of equations. In this paper, we will address the first issue. We will see how under mild assumptions the assemblage procedure may be rewritten using a completely loop-free algorithm. Our approach leads to a small matrix-matrix multiplication for which we may count on highly optimized algorithms.