The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes

A stabilized linearity-preserving scheme for the heterogeneous and anisotropic diffusion problems on polygonal meshes

“…Secondly, we observe that there are some other finite volume schemes that can be casted into the general framework, including the non-linearity-preserving scheme in [12], the linearity-preserving schemes in [23,24], and the nonlinear monotonicity-preserving or extremum-preserving schemes in [27,22]. All the schemes choose the geometric center or the barycenter to be the cell center.…”

“…Li et al [12] Scalar P v K (2.10) (2.14) Equal weight Wu et al [23] Scalar P v K (2.10) (2.14) IW1 Wu and Gao [24] Scalar P v K (2.10) (2.14) EW1 Gao and Wu [8] Tensor P v K (2.10) (2.14) EW2 Yuan and Sheng [27] Tensor P v K (9)-(10) in [22] (2.10) in [27] IW2 Sheng and Yuan [22] Tensor P m K (9)-(10) in [22] (17)- (18) in [22] IW3 In the second step, we consider the problem of finding the expression of the cell matrix A K . Let F K ¼ ðf r;j Þ and X K ¼ ðx i;j Þ be two n K Â 2 matrices whose entries are defined respectively by…”

A finite volume scheme preserving extremum principle for convection–diffusion equations on polygonal meshes

“…In fact, the streamline upwind Petrov-Galerkin method is neither monotone nor monotonicity preserving. Following the idea in [35], a new nonlinear FV scheme for diffusion problem that satisfies the discrete extremum principle has been constructed in [37]. Another approach towards a robust finite element method was developed in [4,5].…”

“…In [7], a nonlinear diamond scheme on unstructured meshes of d -simplices for convectiondiffusion problems is proposed, in which the face gradient is reformulated as a nonlinear average of the one-side gradients by suitably designing solution-dependent weights. Following the idea in [35], a new nonlinear FV scheme for diffusion problem that satisfies the discrete extremum principle has been constructed in [37]. Following the idea in [35], a new nonlinear FV scheme for diffusion problem that satisfies the discrete extremum principle has been constructed in [37].…”

“…We use the method in [37] to eliminate the cell-edge unknowns, that is, We use the method in [37] to eliminate the cell-edge unknowns, that is,…”