研究期間：10108~10207;It is well known that the pair of finite element spaces in the mixed finite element method (FEM) for solving, for example, the Stokes equations must satisfy the so-called inf-sup condition if stable and optimally accurate approximations are desired. This condition prevents the use of standard equal order C0 interpolation spaces for velocity and pressure with respect to the same grid, which are the most attractive from the implementation point of view, or low order pairs such as piecewise linear elements for velocity and piecewise constants for pressure. In order to circumvent the inf-sup condition, a class of so-called stabilized FEMs has been developed and intensively studied for almost thirty years and it is still very attractive today. In this two-year project, we will devote to the development of a new stabilized FEM for solving the system of generalized Stokes equations arising from the time-discretization of the transient Stokes problem and the Brinkman equations modeling the flows through porous media. The system of generalized Stokes equations involves a small kinematic viscosityν (the inverse of the large Reynolds number Re) and a large reaction coefficient σ (the inverse of a small time steptΔ). However, the reaction coefficient σ in the system of Brinkman equations is equal to one and in this case we will focus on the Darcy limiting case, namely, +→0ν. Undisputedly, these coefficients will affect the stability and accuracy of the resulting stabilized finite element solutions. We will first derive the error estimates of the stabilized finite element approximation of the velocity field and the pressure in L2 and H1 norms. We will also explicitly establish the dependence of the error bounds on the viscosity, the reaction coefficient and the mesh size. Based on these error estimates, we will study the adaptive computation and related topics of the newly proposed stabilized FEM. Finally, a series of numerical experiments will be performed, and we will compare numerically the effectiveness of the newly proposed stabilized FEM with some existing stabilization methods in the literature.
