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Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R. The Gradient Discretisation Method

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Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R. The Gradient Discretisation Method
Springer, 2018. — 511 p.
This monograph presents the Gradient Discretisation Method (GDM), which is a unified convergence analysis framework for numerical methods for elliptic and parabolic partial differential equations. The results obtained by the GDM cover both stationary and transient models; error estimates are provided for linear (and some non-linear) equations, and convergence is established for a wide range of fully non-linear models (e.g. Leray–Lions equations and degenerate parabolic equations such as the Stefan or Richards models). The GDM applies to a diverse range of methods, both classical (conforming, non-conforming, mixed finite elements, discontinuous Galerkin) and modern (mimetic finite differences, hybrid and mixed finite volume, MPFA-O finite volume), some of which can be built on very general meshes.
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