Moving Mesh Methods in Multiple Dimensions Based on Harmonic Maps

In practice, there are three types of adaptive methods using the finite element approach, namely the h-method, p-method, and r-method. In the h-method, the overall method contains two parts, a solution algorithm and a mesh selection algorithm. These two parts are independent of each other in the sense that the change of the PDEs will affect the first part only. However, in some of the existing versions of the r-method (also known as the moving mesh method), these two parts are strongly associated with each other and as a result any change of the PDEs will result in the rewriting of the whole code. In this work, we will propose a moving mesh method which also contains two parts, a solution algorithm and a mesh-redistribution algorithm. Our efforts are to keep the advantages of the r-method (e.g., keep the number of nodes unchanged) and of the h-method (e.g., the two parts in the code are independent). A framework for adaptive meshes based on the Hamilton–Schoen–Yau theory was proposed by Dvinsky. In this work, we will extend Dvinskyʹs method to provide an efficient solver for the mesh-redistribution algorithm. The key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure. Each iteration step is to move the mesh closer to the harmonic map. This procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration. The numerical schemes are applied to a number of test problems in two dimensions. It is observed that the mesh-redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi-dimensional computations.