The Finite Volume Scheme Preserving Extremum Principle For Diffusion Equations On Tetrahedral Meshes

In this paper, we construct a nonlinear finite volume scheme for diffu-sion equations on tetrahedral meshes and prove that the scheme satisfies the discrete extremum principle.The discrete extremum principle is an important property on the numer-ical computing of diffusion equation. Constructing a diffusion scheme which satisfies the discrete extremum principle is a difficult problem on arbitrary polygonal meshes. The extremum principle is also one of the key requirements to discretization schemes. The discrete extremum principle can ensure that there is no spurious oscillations for the numerical solution.In this paper, a finite volume scheme which satisfies the discrete extremum principle is constructed on tetrahedral meshes. There are only cell-center un-knowns. In order to reduce the computational complexity, the auxiliary un-knowns are eliminated by integral interpolation. The method ensures the ac-curacy of the scheme and guarantees the discrete extremum principle. Numer-ical results are presented to show how our scheme works for discrete extremum principle on uniform and random tetrahedral meshes. And numerical results show that the scheme is robust and effective.The innovation and difficulty of this paper are:the calculation is large because of the three-dimensional diffusion problem. How to reduce the number of nonzero elements in the coefficient matrix is a difficult problem. The cell-face unknowns are eliminated by locally approximating them with surrounding cell-centered unknowns. In order to satisfy the discrete extremum principle, the coefficients of cell-center unknowns are nonnegative and the sum is 1.