The characteristic mixed finite element method and analysis for three-dimensional moving boundary value problem
Starting from the not everyday observation that ``the famous mathematician\dots and others have published famous papers on the characteristic mixed finite element method'', the author considers the problem of determining Darcy velocity, pressure and a concentration (all connected with transport of oil and gas in a porous medium) under certain boundary conditions in a parallelepiped the boundary of which moves according to a given time-dependent law. The equations for velocity and pressure contain the time only as a parameter whereas the concentration is the solution of a convection-diffusion equation. The problem is transformed to a fixed domain and rewritten into weak form as a saddle point problem combined with an evolution equation. Then a mixed finite element method for the saddle point part and a purely implicit approximation of the evolution equation are shown to converge assuming that the time step is sufficiently small (as compared to the spatial steps), that the exact solution is sufficiently smooth and the (Raviart-Thomas) elements are at least of first order.
characteristic mixed finite element method
scientific article; zbMATH DE number 908304