Application of Moving Mesh Methods for the Solution of Partial Differential Equations

Abstract

This thesis concerns the application of moving mesh methods for the numerical solution of partial differential equations (PDEs) on two-dimensional domains. Recent developments in this field have shown wide applicability for resolving fine scale features such as shocks and singularities in nonlinear PDEs.

As first contribution, we couple a moving mesh PDE and a finite element (FE) method for the solution of the linear advection equation. This scheme also includes a mass-conservative projection operator that avoids any interpolation procedure as used in the standard iterative (rezoning) method. This property is essential for many simulations in the field of computational fluid dynamics (CFD), and prevents the approximate solution from being polluted by the addition of numerical diffusion.

The second contribution of this thesis is related to elliptic PDEs in non-convex domains, discretised with the symmetric interior penalty discontinuous Galerkin (SIP-dG) method. The analytical solution exhibits a singular behaviour at the re-entrant corners, and standard numerical schemes on regular grids are not able to accurately resolve the corner singularities. We will show that both an ℎ-adaptive and 𝑟-adaptive approach can be used to increase considerably the accuracy of the solution. For the first method, we derive a new a-posteriori error estimator using the dual formulation of the original problem.

Different moving mesh methods are tested for 𝑟-adaptivity, and that one based on the Optimal Transport (OT) strategy yields the best accuracy, comparable to the ℎ-refinement method but more computationally efficient. Furthermore, we prove that the local quality measure of the OT mesh is independent on its resolution and cell location. Finally, we provide numerical evidence of the link between the two adaptive methods by showing that the a-posteriori estimator used for ℎ-adaptivity is equidistributed on the OT mesh near the re-entrant corner.

The third contribution proposes an alternative way to solve a model elliptic PDE via deep learning (DL). Under this framework the problem is formulated as the minimisation of a loss function, which encodes desirable properties of the PDE. We show that a neural network with loss defined as the energy functional of the equation yields accurate solution, provided that the training points are chosen according to the equidistribution condition. We link to OT meshes obtained earlier in the thesis. The main result of this work suggests that numerical methods (e.g. used to derive the OT mesh) must be endowed with certain desirable features to ensure compatibility with DL procedures and improve the training process of the neural network.

Date of Award	29 Mar 2023
Original language	English
Awarding Institution	University of Bath
Supervisor	Chris Budd (Supervisor) & Tristan Pryer (Supervisor)