Moving Mesh Method And Its Application

A variety of physical and engineering problems develop dynamically singular or nearly singular solutions in fairly localized regions. In these problems, we are only interested in high resolutions in fairly small solution domain. With uniform meshes, the amount of computational time is too large to enable us to obtain useful numerical approximations, particular in multi-dimensions. Therefore, developing effective and robust moving mesh methods for these problems becomes necessary. Successful implementation of the adaptive strategy can increase the accuracy of the numerical approximations and also decrease the computational cost. In this thesis, we will investigate a class of moving mesh method which is based on harmonic mapping. A logical (or computational) domain is used as a reference and the mesh moving is implemented according to an appropriate domain transformation. With some traditional moving mesh methods, the meshes in physical domain may be tangled. To avoid this, the transformations are constructed based on harmonic mapping. A good feature of the adaptive methods based on harmonic mapping is that existence, uniqueness and non-singularity for the continuous map can be guaranteed from the theory of harmonic maps. Such theoretical guarantees are rare in the field of adaptive mesh generation.In practice, there are three types of adaptive methods using finite element approach, namely the h-method, p-method, and r-method (i.e. moving mesh method). In the h-method, the overall method contains two parts, i.e. a solution algorithm and a mesh selection algorithm. These two parts are independent in the sense that the change of the underlying partial differential equations (PDEs) will affect the first part only. However, in some of the existing r-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 for the r-method (e.g., keep the number of nodes unchanged) and for the h-method (e.g., the two parts in the code are independent). The mesh-moving is a procedure of iteration to construct the harmonic map between the physical mesh and the logical mesh. Each iteration step is to move the mesh closer to the harmonic map. A new scheme to interpolate the approximate solution from the old mesh into the new mesh is designed. The numerical schemes are applied to a number of test problems in two- and three-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 the multi-dimension computations. There have been very few moving mesh results for three-dimensional problems. One of the main difficulties in dealing with the 3D problems is the treatment of the boundary grid re-distribution. In 2D, this difficulty can be handled by solving 1-D moving mesh equations on boundaries. However, the extension of this boundary grid redistribution technique to 3D is very difficult. In order to handle this problem, we will solve a constrained optimization problem that links the interior and boundary points as whole. It turns out that under certain conditions this scheme can be implemented in three dimensions. The mesh generated by the new scheme has higher quality than that generated by moving mesh methods with fixed boundaries.We also applied the moving mesh method to some variational inequality and elliptical control problems. The key idea is to construct the monitor function by using appropriate a posterior error estimates. At the heart of any adaptive finite element refinement schemes are some appropriate a posteriori error estimators. The decision of whether further refinement of meshes is necessary is based on the estimate of the discretization error. If further refinement is to be performed then the a posteriori error estimators are used as a guide as to show the refinement might be accomplished most efficiently. Now the natural question is can we use them as monitor functions in the moving mesh methods? One of the objectives of this thesis is to address the above question. It is shown that the moving mesh methods with appropriate monitor functions can effectively solve elliptic obstacle problems and elliptic control problems.The thesis is organized as follows. In Chapter 0, we list some basic results to be used in the thesis. Chapter 1 provides some introduction to adaptive finite element method. In Chapter 2, the moving mesh method based on harmonic mapping will be investigated. Chapter 3 is devoted to the discussions of the moving mesh method with boundary redistribution. In Chapter 4, we consider the application of the moving mesh methods for variational inequality and elliptic optimal control problems, with particular attention to the construction of the monitor functions. Chapter 5 presents some numerical computations for partial differential equations, while Chapter 6 shows the numerical computations for variational inequality and elliptic optimal control problems. In Chapter 7, we present numerical results for two-dimensional problems with boundary grid-redistribution. Numerical results for three-dimensional computations are presented in Chapter 8. Some consluding discussions are given in the final chapter. In appendix A, a symmetric error estimate for the moving mesh method will be briefly demonstrated.