A monotone combination scheme of diffusion equations on polygonal meshes

We present a novel monotone scheme which is a combination of linear scheme and nonlinear monotone scheme for solving diffusion problems on general polygonal meshes. It will be called as a combination scheme and consists of two steps. Firstly, a second-order accurate linear scheme is used to obtain an approximate solution. Secondly, a nonlinear monotone scheme is used to solve the diffusion equation, where the unknowns in the nonlinear coefficient of the nonlinear monotone scheme are taken as the approximate solution of the linear scheme above, i.e., a linearized monotone scheme is obtained. So the combination scheme does not require nonlinear iterations for solving linear diffusion problems, moreover, it benefits from the accuracy and efficiency of the linear scheme, as well as the monotonicity of nonlinear monotone scheme. We also analyze some properties satisfied by the combination scheme, such as conservation, stability, monotonicity and convergence. Numerical results are presented to show the performance of the monotone combination scheme on distorted meshes, especially some numerical comparisons among our combination scheme with some existing linear and nonlinear monotone schemes are given.