Solution of Parabolic Equations by Backward Euler-Mixed Finite Element Methods on a Dynamically Changing Mesh

We develop and analyze methods based on combining the lowest-order mixed finite element method with backward Euler time discretization for the solution of diffusion problems on dynamically changing meshes. The methods developed are shown to preserve the optimal rate error estimates that are well known for static meshes. The novel aspect of the scheme is the construction of a linear approximation to the solution, which is used in projecting the solution from one mesh to another. Extensions to advection-diffusion equations are discussed, where the advection is handled by upwinding. Numerical results validating the theory are also presented.