Studies On Finite Volume Element Methods For Interface Problems

The finite volume element methods have been one class of the most commonly used numerical methods for solving partial differential equations. Generally speaking, finite volume element methods not only are flexible in handling complicated domain geometries and boundary conditions, but also could keep the conservation property of some certain physical quantities. Therefore, a lot of relevant works have been devel-oped. But nearly all of them are concerned with non-interface problems. However, there are a mass of problems whose physical domains are filled with multi-material in practice. People refer to them as interface problems. The numerical solutions of inter-face problems are more challenging and significative. In this paper, we study the finite volume element methods for three kinds of interface problemsFirst, we consider the Darcy flow model for the pressure and velocity whose com-putational domain is a multi-block domain structure. The diffusion tensors may be discontinuous across the interfaces between different sub-domains. There are two typi-cal examples in subsurface porous medium applications. One is the modeling of faults, which are natural discontinuities in material properties. The other is the modeling of wells in which more accurate numerical gradients are desired. In the numerical sim-ulation of such problems, each sub-domain is independently covered by a conforming grid. Thus, grids do not match on the interfaces between different sub-domains. The non-matching grids make the numerical discretization and the corresponding theoretical analysis of such problems more difficult to carry out. In this paper, we study the mixed finite volume element methods for the Darcy flow model on non-matching grids.Secondly, we consider two-dimensional three-temperature (2-D 3-T) radiation d-iffusion equations.2-D 3-T radiation diffusion equations belong to non-equilibrium radiation diffusion equations that are widely used to simulate problems in inertial con-finement fusion experiments, magnetic confinement fusion experiments, astrophysics problems, and so on.2-D 3-T radiation diffusion equations are widely used to approx- imately describe the evolution of radiation energy within a multi-material system and explain the exchange of energy among electrons, ions and photons. Their highly non-linear, strong discontinuous and tightly coupled phenomena always make the numeri-cal solution of such equations extremely challenging. There are two key requirements desired to be satisfied in the numerical simulation of such problems. One is energy conservation property, the other is monotonicity, i.e. preserving positivity of analytical solutions. Besides, the computation is too time-consuming, so how to reduce the cost is also an issue we need to consider. In this paper, we devise two finite volume ele-ment schemes and a mesh adaptation algorithm for solving 2-D 3-T radiation diffusion equations.At last, we consider the immersed interface problems. The computational domain is separated into two sub-domains by a sufficiently smooth interface. The diffusion coefficient is continuous on each sub-domain, but discontinuous across the interface. The mesh partition is independent of the interface. Thus, there would be some elements cut through by this interface. If using the classical conventional finite element methods to solve this interface problem, it can not achieve optimal convergence rates. In order to overcome this obstacle, a kind of new finite element spaces, i.e., the immersed finite element spaces is proposed. Then, the immersed finite element methods, i.e., the finite element methods based on the immersed finite element spaces, have nearly optimal convergent rates. Although this kind of methods have been developed rapidly, their theoretical analysis is extremely difficult and still open issues until now. In this paper, we extend the immersed finite element spaces to finite volume element methods and construct a new numerical scheme by adding two penalty terms.This paper includes six chapters. We draw the conclusion in the last chapter, and the other chapters are organized as followed.In the first chapter, we introduce the background and development of our research briefly, i.e., mixed methods for Darcy flux model on non-matching grids, numerical solutions of 2-D 3-T radiation diffusion equations and immersed finite volume element methods.In the second chapter, we consider the Darcy flux model on non-matching grids, i.e., the first interface problem that is mentioned above. Each sub-domain is covered by a independent triangular grids, and by connecting the barycenter and the three nodes in each triangular we get the dual partition. On each sub-domain, we take the lowest-order Raviart-Thomas space and piecewise constant space to approximate the velocity and pressure respectively and employ the standard mixed finite volume methods to dis-cretize original equations. Because the grids are non-matching on each interface, the approximate velocity space no longer satisfies the continuity of flux and there would be an additional term related to the trace of pressure. In this chapter, we introduce a mortar finite element space on the interface to approximate the pressure and add an interface condition to unite the sub-domain problems. The resulted scheme is called mortar mixed finite volume element method. We prove its optimal convergence rate in the L2-norm by theoretical analysis and numerical examples.In the third chapter, we still consider the Darcy flux model on non-matching grids, i.e., the first interface problem that is mentioned above. The mesh partition, dual par-tition and approximate velocity space and pressure space are the same as the second chapter. On each sub-domain, we still employ the standard mixed finite volume meth-ods to discretize original equations, but the way how to deal with the non-matching interface is different. In this chapter, the Robin type conditions are imposed weakly on the non-matching interfaces by using double-valued Lagrange multipliers to approxi-mate the trace of the pressure. The resulted scheme is called non-mortar mixed finite volume element method. In addition, we prove its optimal convergence rate in the L2-norm by theoretical analysis and numerical examples.In the forth chapter, we consider 2-D 3-T radiation diffusion equations, i.e., the second interface problem that is mentioned above. There are two key requirements desired to be satisfied in the numerical simulation of such problems. One is energy conservation property, the other is monotonicity, i.e., preserving positivity of analytical solutions. In this paper, from the energy conservative form of the original system, we adopt some reasonable numerical integral formulas and approximate methods to deal with nonlinear terms and discontinuous coefficients. By various numerical integral for-mulas, we construct two kinds of conservative finite volume element schemes. From the analysis of monotonicity and the numerical experiments, we know that the first one is monotone on many kinds of meshes, and the corresponding restrictive conditions of meshes are derived. However, the other one is nearly impossible to monotone and a large part of the numerical solutions would be negative at the start of actual calculation. Therefore, this scheme is usually considered not available. But we specially design two repair techniques for 2-D 3-T radiation diffusion equations, including the global repair technique and cutoff method, to overcome this defect. From the numerical results, we find that both of them could achieve good computational efficiency. Additionally, a spacial mesh adaptation algorithm based on the residual-type posteriori error estimator is also proposed, by which the cost can be reduced observably.In the fifth chapter, we construct a new immersed finite volume element method for the third interface problem that is mentioned above. Let elements cut through by the interface be interface elements, whereas the other elements be non-interface elements. The immersed finite element space is defined as follows:on interface elements it is a piecewise linear polynomial and on non-interface elements it is a linear polynomial. If taking the immersed finite element space as the trial space in the standard finite volume element method, the resulted scheme is called immersed finite volume element method. In this chapter, we modify this method by penalizing the jump conditions on the solution value as well as the flux across the interface and the edges intersecting with the inter-face in the bilinear form. The numerical example indicates that this modified method can achieve optimal convergence rates in both L2-norm and H1-norm. We provide a rigorous theoretical analysis to show that it can well keep stability even though the jump of the coefficient is taken a large value.