Some Studies On Discontinuous Finite Volume Element Methods

The discontinuous finite volume method is a numerical method developed in the past ten years.It not only has the flexibility,high order of accuracy,easy parallelism of the discontinuous finite element method,maintaining physical conservation,and for some problems,it is simpler than the discontinuous finite element method.However,most of the current research focuses on the format construction of two-dimensional triangular grids.For the discontinuous finite volume method on the general one-dimensional problem and two-dimensional problem of the quadrilateral grid,there are relatively few studies.Therefore,the paper establishes discontinuous finite volume element formats and gives the theoretical analysis.For a type of one-dimensional time-dependent convective diffusion equations,a new discontinuous finite volume element format is established by using the numerical flux introduced in the discontinuous finite element method.The stability analysis of the format is given for periodic boundary value condition.The new format uses piecewise constant functions as test functions,which is simpler than the corresponding discontinuous finite element method.Numerical experiments show that the numerical solution of the linear discontinuous finite volume element converging to exact solution with 2-th order of the L2 norm.The method is also suitable for solving nonlinear parabolic problem with degradation coefficient.In order to avoid the non-physical oscillation problem of numerical solutions,global repair techniques are introduced in numerical calculation.For the second order elliptic equation with full tensor diffusion coefficient,a new format of discontinuous finite volume element based on bilinear element is established in the rectangular grid,which is different from the dual partition in the literature.In the new format,dual grid is constructed by directly connecting the cell to the middle point of the edge.The stability and convergence analysis of the discontinuous finite volume element method are given by choosing proper interpolation operator.The numerical results verify the correctness of the theoretical analysis.These studies are significant for further improving the theoretical research and numerical analysis of the discontinuous finite volume method.