Adaptive interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

In this paper we conduct a priori and a posteriori error analysis of the C0 interior penalty method for Hamilton–Jacobi–Bellman equations, with coefficients that satisfy the Cordes condition. These estimates show the quasi-optimality of the method, and provide one with an adaptive finite element method. In accordance with the proven regularity theory, we only assume that the solution of the Hamilton–Jacobi–Bellman equation belongs to H2

An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing

Moving sharp fronts are an important feature of many mathematical models from physical sciences and cause challenges in numerical computation. In order to obtain accurate solutions, a high resolution of mesh is necessary, which results in high computational cost if a fixed mesh is used. As a solution to this issue, an adaptive mesh method, which is called the moving mesh partial differential equation (MMPDE) method, is described in this work. The MMPDE method has the advantage of adaptively relocating the mesh points to increase the densities around sharp layers of the solutions, without increasing the mesh size. Moreover, this strategy can generate a nonsingular mesh even on non-convex and non-simply connected domains, given that the initial mesh is nonsingular. The focus of this thesis is on the application of the MMPDE method to mathematical models from physical sciences and image segmentation. In particular, this thesis includes the selection of the regularization parameter for the Ambrosio-Tortorelli functional, a simulation of the contact sets in the evolution of the micro-electro mechanical systems, and a numerical study of the flux selectivity in the Poisson-Nernst-Planck model. Sharp interfaces take place in all these three models, bringing interesting features and rich phenomena to study

Surface and bulk moving mesh methods based on equidistribution and alignment

In this dissertation, we first present a new functional for variational mesh generation and adaptation that is formulated by combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter. The functional is shown to be coercive which, when employed with the moving mesh partial differential equation method, allows various theoretical properties to be proved. Numerical examples for bulk meshes demonstrate that the new functional performs comparably to a similar existing functional that is known to work well but contains an additional parameter. Variational mesh adaptation for bulk meshes has been well developed however, surface moving mesh methods are limited. Here, we present a surface moving mesh method for general surfaces with or without explicit parameterization. The development starts with formulating the equidistribution and alignment conditions for surface meshes from which, we establish a meshing energy functional. The moving mesh equation is then defined as the gradient system of the energy functional, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. A selection of two-dimensional and three-dimensional examples are presented

Advanced numerical and statistical techniques to assess erosion and flood risk in coastal zones

Throughout history, coastal zones have been vulnerable to the dual risks of erosion and flooding. With climate change likely to exacerbate these risks in the coming decades, coasts are becoming an ever more critical location on which to apply hydro-morphodynamic models blended with advanced numerical and statistical techniques, to assess risk. We implement a novel depth-averaged hydro-morphodynamic model using a discontinuous Galerkin based finite element discretisation within the coastal ocean model {\em Thetis}. Our model is the first with this discretisation to simulate both bedload and suspended sediment transport, and is validated for test cases in fully wet and wet-dry domains. These test cases show our model is more accurate, efficient and robust than industry-standard models. Additionally, we use our model to implement the first fully flexible and freely available adjoint hydro-morphodynamic model framework which we then successfully use for sensitivity analysis, inversion and calibration of uncertain parameters. Furthermore, we implement the first moving mesh framework with a depth-averaged hydro-morphodynamic model, and show that mesh movement can help resolve the multi-scale issues often present in hydro-morphodynamic problems, improving their accuracy and efficiency. We present the first application of the multilevel Monte Carlo method (MLMC) and multilevel multifidelity Monte Carlo method (MLMF) to industry-standard hydro-morphodynamic models as a tool to quantify uncertainty in erosion and flood risk. We use these methods to estimate expected values and cumulative distributions of variables which are of interest to decision makers. MLMC, and more notably MLMF, significantly reduce computational cost compared to the standard Monte Carlo method whilst retaining the same level of accuracy, enabling in-depth statistical analysis of complex test cases that was previously unfeasible. The comprehensive toolkit of techniques we develop provides a crucial foundation for researchers and stakeholders seeking to assess and mitigate coastal risks in an accurate and efficient manner.Open Acces