We present a finite difference method for mesh generation and adaptation with application to the solution of partial differential equations in curved domains. For mesh generation, we construct a mapping between a fixed rectangular domain and a curved physical domain using a finite difference approximation of the fully nonlinear Monge–Ampere equation solved with a damped Newton’s method. Paired with grid adaptation, this method gives a finite difference approximation for curved geometries that distributes mesh points toward localized interior features such as sharp interfaces using a solution dependent monitor function. The method dynamically resolves fine scale PDE behavior on stationary and evolving curved domains while retaining the simplicity of performing all computations on a static rectangular grid with a fixed number of mesh points and connectivity. We display the efficacy of the methods on a variety of linear and nonlinear PDE examples on convex and select nonconvex curved domains.

In this talk, we introduce an efficient domain decomposition (DD) approach for adaptive mesh generation in two and three spatial dimensions. The computational domain is partitioned into overlapping subdomains, and on each subdomain the adaptive mesh is computed as an image of the unique solution of the \(L^2\) optimal mass transfer problem using parabolic Monge-Ampere (PMA) method. The global adaptive mesh is reconstructed using a technique similar to the Schwarz alternating method. We present several numerical experiments to demonstrate the performance of the proposed domain decomposition parabolic Monge-Ampere moving mesh method (DDPMA). The numerical results indicate that the DDPMA method compare favorably to the PMA method employed on a single domain, in particular when computing adaptive meshes in two and three spatial dimensions.

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, frequency, space, and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. In this talk, we present the solution of the radiative transfer equation using an adaptive moving mesh DG method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Numerical examples are presented to demonstrate the mesh adaptation ability, accuracy, and efficiency of the method. This talk is based on joint work with Juan Cheng, Weizhang Huang, and Jianxian Qiu.

In this talk, we introduce adaptive techniques for solving PDEs called moving mesh methods that allow the mesh points to tend toward the regions where the solution varies and away from the regions where the solution exhibits little change. The mesh points evolve alongside the physical solution and the motion of the mesh is controlled by another PDE with an associated monitor function. One issue with this coupled system of PDEs is that steep gradients or large function values in the physical problem will cause similar characteristics in the mesh solution. We show that for certain choices of the monitor function, we can balance this undesirable behavior between the mesh equation and the physical equation, giving a system that is overall more efficient to compute. Applications presented include a phase transition model and a reaction problem that blows up in finite time.

The Radon transform and its inverse are important tools in producing images from medical devices such as CT (Computerized Tomography) scanners. Chebyshev spectral methods are an accurate and efficient tool for numerical approximating solutions to partial differential equations. In this work we present a method that combines the Radon transform and Chebyshev spectral methods to produce an accurate and efficient framework for solving multidimensional linear hyperbolic partial differential equations (PDEs). The key innovation is that the Radon transform allows us to only solve a set of decoupled one-dimensional PDEs along each Radon transform direction, while the multidimensional nature of the solution is recovered in the Radon inversion process. We apply the resulting method to a series of examples involving the wave equation and the PN approximation of the radiative transfer equation.

In this talk we present two approximation methods for computing eigenfrequencies and eigenmodes of large-scale nonlinear eigenvalue problems resulting from boundary element method (BEM) solutions of some types of acoustic eigenvalue problems in three-dimensional space. We first approximate the resulting boundary element matrix by a high accuracy rational approximation using the Cauchy integral formula or the Chebyshev interpolation. Then a Rayleigh-Ritz procedure, which is suitable for parallelization is developed for both the Cauchy and the Chebyshev approximation methods when dealing with large-scale practical applications. The performance of the proposed methods is illustrated with a variety of benchmark examples and large-scale industrial applications. This is joint work with M. El-Guide and Y. Saad.

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, 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. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed based on the current mesh. A selection of two- and three-dimensional examples are presented.

The phase-field approach for numerical simulation of initiation and propagation of brittle fracture will be presented. Challenges of the approach, including non-smoothness of the energy functional, violation of fracture boundary conditions, and the need for mesh adaptation, and possible remedies for these challenges will be discussed. In particular, a moving mesh finite element method will be presented for the numerical solution of a phase-field model for brittle fracture.