The Finite Volume Element Method For Two Classes Of Interface Problems

This thesis is devoted to studying the finite volume element method for two class-es of interface problems. The contents are arranged into three parts.In Chapter 1, firstly, we give an introduction of some immersed methods about the interface problems, and illustrate the purpose of developing the immersed finite volume element method. Then, we introduce 1D dual-phase lag （DPL） heat con-duction equations with the interface, the high-order compact finite volume element method and the Pade compact finite volume method.In Chapter 2, Part 1 discusses an immersed finite volume element method for the Poisson equation with an interface. Using the source removal technique, the interface problem with nonhomogeneous jump conditions is transformed into a new one with homogeneous jump conditions. Then the term related to the nonhomogeneous jump conditions is removed to the right hand side of the equation. Then, the bilinear basis function is the usual finite element basis function with a uniform rectangular parti-tion. Two numerical examples are presented to demonstrate the efficiency and prac-ticability of the new method. Further, Part 2 discusses the immersed finite volume element method to solve 2D elliptic interface problems with discontinuous variable coefficients that has a finite jump across an interface. The solution and the flux may also have a finite jump across the interface which increase the numerical difficulties. Using the source removal technique, an equivalent elliptic interface problem with ho-mogeneous jump conditions is obtained. The nodal basis functions are constructed as piecewise polynomial functions to satisfy the homogeneous jump conditions near the interface. And the usual finite element nodal basis functions are applied away from the interface. There are some singularities with the rectangular grids. So the trian-gular grids are applied. The resulting linear problem is simple and easy to solve. A proof of the error estimate in the energy norm is given. Two numerical experiments demonstrate the errors near the interface and the global errors of the proposed method with the usual O（h2） in the L2, the L∞ norms, and O（h） in the H1 norm.In Chapter 3. a high-order compact finite volume element method is presented for 1D dual-phase lag （DPL） heat conduction equations with the interface. The result- ing coefficient matrix is three-diagonal except for the schemes of the interface point which have some good symmetrical and diagonal properties. This high-order method is helpful to analyze and study nano heat conduction with this equation in relative coarse grid. We apply the discrete energy method to give the error estimate in the L2 norm with the convergence order O(△t²+h^3.5). Finally, numerical examples are provided to show the effectiveness and feasibility of this method.The construction of this high-order method involves the back substitution of the equations. Once we encounter the equations with many variables, this method may become impractical. The following research will focus on a convenient high-order method to solve multi-dimensional the interface problems. Subsequently, we simply discuss a Pade high-order compact finite volume method for the interface problems. And the method allows the solution and its derivatives to have 4th-order accurate.In Chapter 4,the major conclusions of this thesis and problems to be further con-sidered are listed.