Collaborators
Projects
- Precondition viscoplastic problems by introducing an elastic term and using TSPSEUDO to timestep it out as a nonlinear preconditioner for the SNES
- Combining discontinuous (dG) and continuous Galerkin (cG) methods
Motivation
In magma dynamics, we have a complicated, nonlinear set of algebraic constraints which ensure conservation of mass and momentum for the flow of an incompressible liquid inside of a compressible solid. The solid is charactrized by a porosity which determines how much liquid is present (I believe we are always fully saturated). The porosity is advected by the solid velocity and we must do this with no diffusion and little dispersion since it can support shocks and solitary waves.
We would like to use cG to discretize the algebraic constraints since they are elliptic and we have optimal preconditioners for them (Rhebergen, et.al.). We would like to use an accurate, conservative scheme for advancing the porosity advection, so we use dG.
Background
There has been work combining cG and dG over different parts of the domain by both Riviere and Dawson. There is a suggestive paper by Dawson and Proft looking at cG+dG for shallow water. We might be able to follow this one.
There is work by Li and Riviere looking at miscible flow where they prove stability by adding a term so that the operator is skew-symmetric (thus the eigenvalues are pure imaginary and you get damping from another part of the problem).
We have implemented a higher-order Lagrange for the constraints, and FV for the porosity.
Project
- Do MMS with discontinuous functions by calculating weak derivatives to get the projection on the FEM space, and then represent this by some measure (I think). Short paper.