Two-grid Eulerian-lagrangian Localized Adjoint And Immersed Finite Element Methods For Two-phase Flow Problems

Many problems arise in the numerical simulation of fluid flow processes within porous media in petroleum reservoir simulation and in subsurface contaminant trans-port and remediation.The mathematical models used to describe these complex flow processes are coupled systems of time-dependent nonlinear partial differential equa-tions.Due to the nonlinearity and couplings of these equations,the moving steep fronts present in the solution of these partial differential equations,and the enormous size of field-scale application,the numerical treatment of these systems often encoun-ters severe difficulties.Standard finite difference or finite element methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion a-long with spurious grid-orientation effect.For immiscible two-phase flow problems,there is usually an interface.Due to the discontinuity of the coefficients between the interfaces,the complex geometric structure of the interface and the low global regularity,it is very challenging to solve the interface problem efficiently.As the computation domain of two-phase flow problem is very large,we are going to construct efficient and fast algorithms to improve the computational efficiency of these problems.In this paper,we study two-grid Eulerian-Lagrangian local adjoint methods(ELLAM)for two-phase problems and two-grid immersed finite element methods for non-linear interface problems.The main researches are as follows:First of all,we study two-grid method for incompressible two-phase miscible dis-placement problems by mixed finite element and ELLAM schems.In this section,we consider the diffusion term only with the molecular diffusion,present mixed finite element and ELLAM discretization and prove the L4 norm error estimate of the dis-cretization scheme.Then,we propose a two-grid algorithm based on Newton iteration and derive the error estimates.Finally,numerical examples are given to verify the efficiency of two-grid algorithm.Secondly,we propose a two-grid algorithm for incompressible two-phase miscible displacement problems with dispersion term by mixed finite element and ELLAM algorithm.In this part,we present mixed finite element and ELLAM discretization for incompressible two-phase miscible displacement problems with dispersion term and prove the L4 norm error estimate of the coupled scheme.Then,we present a two-grid method based on Newton iteration and get the convergence results.Finally,numerical examples are given to verify the efficiency of two-grid algorithm.Furthermore,we study two-grid immersed finite element methods for non-linear elliptic interface problems.We first present immersed finite element discretization scheme for non-linear elliptic interface problems and prove the error estimates in H1 and Lp norm.Then,we investigate two-grid methods based on Newton iteration and derive the error estimates.Moreover,we consider the non-linear elliptic interface problems with nonhomogeneous jump condition.A novel two-grid algorithm has been investigated and analyzed.Numerical experiments are presented to support the theoretical analysis and indicate that two-grid algorithms are very effective.Last but not least,we study two-grid method for non-linear elliptic interface problems by partially penalized immersed finite element method.Extra stabilization terms are introduced at interface edges for penalizing the discontinuity in immersed finite element(IFE)functions.Optimal error estimates in both H1 and Lp norms are obtained for the immersed finite element discretizations.To linearize the immersed finite element method equations,two-grid algorithm based on some Newton iteration approach is applied.It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h1/4).Numerical examples are given to verify the efficiency of two-grid algorithm.