Minimal Residual Finite Element Methods for First-Order Partial Differential Equations

Abstract

This work addresses numerical challenges in solving first-order partial differential equations (PDEs) with discontinuous solutions. It focuses on the least-squares finite element method (LS-FEM), which suffers from spurious oscillations near discontinuities. The second part of the thesis focuses on minimal residual finite element methods (MinRes FEM), which avoids oscillations but produces inaccurate residual functions, limiting the effectiveness of the method.

To reduce these oscillations, we propose a modified LS-FEM that incorporates total variation (TV) regularisation. This approach penalises nonphysical oscillations while preserving sharp gradients. The resulting convex but non-smooth minimisation problem is solved using a primal-dual algorithm. We establish a link to algebraic solvers that employ $L^1$-regularisation and offer a quasi-optimal analysis of the errors introduced by both the regularisation and the discretisation. Numerical results for PDE problems confirm that the TV-regularised LS-FEM reduces oscillations and improves accuracy, especially in capturing discontinuities.

To improve MinRes FEM, we introduce a hybrid method that uses a neural network to approximate the residual and finite element discretisation for the solution. This method combines the flexibility of neural networks with the physical consistency of finite elements. It is formulated as a saddle point problem and solved through an iterative scheme. Moreover, we establish quasi-optimal a priori and a posteriori error estimates under appropriate regularity assumptions. Numerical experiments confirm that this hybrid approach produces stable, oscillation-free, and accurate solutions. These results demonstrate the effectiveness of the neural network in approximating the residual.