A finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes

•A linear FVE scheme is designed for quadrilateral meshes with even concave cells.•A generic monotone correction is suggested to create a nonlinear monotone scheme.•The two schemes are for discontinuous anisotropic coefficients and concave meshes.•The existence and monotonicity of the solution are proved for the monotone scheme.•The two schemes reach nearly optimal convergence rates on distorted meshes.