Monotonicity Correction For Low Order Element Finite Volume Methods

We apply the monotonicity correction to low-order element finite volume methods,which include linear element on triangular meshes and bilinear element on quadrilateral meshes,for anisotropic diffusion problem.The new schemes have monotonicity that the numerical solutions are non-negative.When we formulate the linear element finite volume schemes,we need to calculate line integrals on the boundary line segment of the dual el-ement,and regard these line integrals as numerical flux.These numerical flux contain the two-points flux structure.Therefore,we can apply nonlinear monotonic correction to these numerical flux,then we obtain a corrected low-order element finite volume schemes whose stiffness matrix is M-matrix.Because inverse of M-matrix is nonnegative,when the source term is non-negative,the nonnegative properties of numerical solutions are guaranteed.Fur-thermore,for the time dependent problem,we also apply monotone correction to time deriva-tive term.Numerical experiments show that the corrected low-order element finite volume schemes have monotonicity,the numerical solutions have nonnegativity and maintains the convergence order of the original schemes,which are second order in L~2 norm and first order in H~1 norm.