The gradient discretisation method
This monograph focuses on the Gradient Discretisation Method, which is introduced as a unified convergence analysis framework for numerical methods for elliptic and parabolic partial differential equations. Several examples of numerical methods are shown to fall into the range of the discrete gradient discretization methods: The GDM applies e.g. to both classical (conforming, non-conforming, mixed finite elements, discontinuous Galerkin) and modern (mimetic finite differences, hybrid and mixed finite volume) discretization methods. The convergence results are obtained by the GDM for both stationary and transient models. An error estimates are provided for linear equations, and several nonlinear equations are investigated as well. The convergence is established for a wide range of fully non-linear models as Leray-Lions equations and degenerate parabolic equations such as the Stefan or Richards model.
parabolic partial differential equations
discrete Aubin-Simon compactness theorems
scientific article; zbMATH DE number 6897811