Finite Volume Scheme For Diffusion Equation On Arbitrary Polygon

In this paper,we construct a new element centered finite volume scheme for the 2D steady and unsteady diffusion equations.To discrete the diffusion operator in space,we employ the mesh vertex unknowns as auxiliary ones,by solving an underdetermined linear system of equations,the auxiliary interpolation point information is replaced by the center point information of the mesh element.Finally,the discretization scheme containing only the unknown quantity of the mesh element center is obtained.Compared with the classical arithmetic average weighting,inverse distance weighting and bilinear interpolation weighting,the new scheme not only satisfies the local conservation condition and linear accuracy criterion,but also makes the new scheme suitable for arbitrary polygonal meshes.For the unsteady diffusion equations,we discrete the time derivative term by Backward Euler scheme.The stability and convergence of the algorithm are analyzed theoretically.The numerical experiments are carried out on random quadrilateral mesh,Shestakov mesh,triangular mesh,random triangular mesh and polygonal mesh respectively,For the case that the diffusion coefficient is continuous constant,discontinuous constant and tensor form,it is found that the error of the new scheme can reach the second-order convergence rate.For the nonlinear diffusion equation whose diffusion coefficient depends on unknown variables,the new scheme still shows the optimal convergence,Furthermore,the robustness of the new element centered finite volume scheme is verified.