We apply the monotonicity correction to thefinite element method for the anisotropic diffusion problems,including linear and quadraticfinite elements on triangular meshes.When formulating thefinite element schemes,we ...We apply the monotonicity correction to thefinite element method for the anisotropic diffusion problems,including linear and quadraticfinite elements on triangular meshes.When formulating thefinite element schemes,we need to calculate the integrals on every triangular element,whose results are the linear combination of the two-point pairs.Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs.We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of thefinite element equations.Finally,the original stiffness matrix can be transformed into a nonlinear M-matrix,and the corrected schemes have the positivity-preserving property.We also give the monotonicity correction to the time derivative term for the time-dependent problems.Numerical experiments show that the correctedfinite element method has monotonicity and maintains the convergence order of the original schemes in H1-norm and L2-norm,respectively.展开更多
基金supported by the Science Challenge Project(No.TZ2016002)the National Science Foundation of China(No.12071177,No.11971069).
文摘We apply the monotonicity correction to thefinite element method for the anisotropic diffusion problems,including linear and quadraticfinite elements on triangular meshes.When formulating thefinite element schemes,we need to calculate the integrals on every triangular element,whose results are the linear combination of the two-point pairs.Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs.We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of thefinite element equations.Finally,the original stiffness matrix can be transformed into a nonlinear M-matrix,and the corrected schemes have the positivity-preserving property.We also give the monotonicity correction to the time derivative term for the time-dependent problems.Numerical experiments show that the correctedfinite element method has monotonicity and maintains the convergence order of the original schemes in H1-norm and L2-norm,respectively.
