On the monotonicity of high order discrete Laplacian

The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, im5 plies discrete maximum principle, which is in general not true for high order schemes on unstructured meshes. But on structured meshes, it is possible to have high order accurate monotone schemes. We first review previously known high order accurate inverse positive schemes, all of which are fourth order accurate with proven monotonicity on uniform meshes. Then we discuss the monotonicity of a fourth order variational difference scheme on quasi-uniform meshes and prove the inverse positivity of a fifth order accurate variational difference scheme on a uniform mesh.