Unconditional Convergence and Optimal Error Estimates of a Galerkin-Mixed FEM for Incompressible Miscible Flow in Porous Media

Abstract

In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal L 2 error estimates hold without any time-step (convergence) conditions, while all previous works require certain time-step restrictions. Our theoretical results provide a new understanding on commonly-used linearized schemes. The proof is based on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

“…In this section, we provide a primary error estimates for the linearized BDF-Galerkin FEMs (2.7)-(2.8) unconditionally by Li-Sun error splitting technique [23,24]. For n = 3, 4, .…”

“…However, the time step restriction condition of linearized schemes arising from error analysis is always a crucial issue. We refer to [14,20,24,29] for works on some typical nonlinear parabolic problems. Because of difficulties in obtaining the boundedness of the numerical solution in certain strong norms, which is an essential condition for error analysis of nonlinear problems, most previous works require certain time step restrictions.…”

“…Our work in this paper was inspired by [23,24], where a new approach was proposed by Li and Sun to get unconditional stability and optimal error analysis of a semi-implicit backward Euler scheme with a linear FEM for the time-dependent thermistor equations and a linearized Galerkin-mixed FEM for porous media flow problem, respectively. The new approach was based on a temporal-spatial error splitting technique by introducing a corresponding timediscrete system.…”

“…Thus, with the help of energy method argument, the results obtained for the first order schemes in [13,23] and the second order scheme in [22] can be extended to higher order BDF type schemes. Combining the Li-Sun error splitting argument [23,24] and the telescope formula for the BDF temporal discretization operator [25], we are able to derive a primary error estimate unconditionally which directly implies uniform boundedness of the FEM solutions in certain strong norms. Then we apply a traditional approach to obtain an optimal L 2 error estimate for r -th order Lagrange element methods (r ≥ 1) unconditionally.…”

Convergence of a linearized second-order BDF-FEM for nonlinear parabolic interface problems

“…A challenge is to prove that the numerical solution is also pointwise bounded. For this purpose, we use the error splitting technique introduced in [18,19]. We shall see that the nonlinear interface condition also imposes difficulties on the analysis of high-order time-discretization schemes such as the second-order BDF.…”