Monotonicity Correction for the Finite Element Method of Anisotropic Diffusion Problems (Communications in Computational Physics, 2022)
We apply the monotonicity correction to the finite element method for theanisotropic diffusion problems, including linear and quadratic finite elements on triangular meshes. When formulating the finite element schemes, we need to calculatethe integrals on every triangular element, whose results are the linear combinationof the two-point pairs. Then we decompose the integral results into the main andremaining parts according to coefficient signs of two-point pairs. We apply the nonlinear correction to the positive remaining parts and move the negative remaining partsto the right side of the finite element equations. Finally, the original stiffness matrixcan be transformed into a nonlinear M-matrix, and the corrected schemes have thepositivity-preserving property. We also give the monotonicity correction to the timederivative term for the time-dependent problems. Numerical experiments show thatthe corrected finite element method has monotonicity and maintains the convergenceorder of the original schemes in $H^1$-norm and $L^2$-norm, respectively.