Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media

摘要

In this paper, we study the unconditional convergence and error estimates ofa Galerkin-mixed FEM with the linearized semi-implicit Euler time-discretescheme for the equations of incompressible miscible flow in porous media. Weprove that the optimal $L^2$ error estimates hold without any time-step(convergence) condition, while all previous works require certain time-stepcondition. Our theoretical results provide a new understanding on commonly-usedlinearized schemes for nonlinear parabolic equations. The proof is based on asplitting of the error function into two parts: the error from the timediscretization of the PDEs and the error from the finite element discretizationof corresponding time-discrete PDEs. The approach used in this paper isapplicable for more general nonlinear parabolic systems and many otherlinearized (semi)-implicit time discretizations.