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Advanced Numerical Methods in the Mathematical Sciences

Konstantin Lipnikov, Los Alamos National Laboratory
The Mimetic Finite Difference and Virtual Element Methods


The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems such as conservation laws, duality and self-adjointness of differential operators, and exact mathematical identities of the vector and tensor calculus. The MFD method works on general polygonal and polyhedral meshes. The virtual element method (VEM) was introduced recently as an evolution of the nodal MFD method.

In the first part of this talk, I'll review common underlying concepts of both methods, with the focus on consistency and stability conditions and efficient computer implementation of the methods. In the second part of the talk, I'll show how flexibility of the MFD framework can be used to design new schemes with special properties for nonlinear evolution problems.

More precisely, I'll explain how upwinding of problem coefficients can be included in mimetic schemes.