دانلود رایگان مقاله برآورد خطای پسینی برای ثبات اجزای محدود مخلوط برای مساله استوکس

1. Introduction

It is well known that the low-order mixed finite element P1/P0 (linear velocity, constant pressure) and P1/P1 (linear velocity and pressure) pairs do not satisfy the inf–sup condition (see, e.g., [1]). Since the low-order pairs remain a popular practical choice in mixed finite element approximation of incompressible models, several stabilized finite element methods have been developed in last two decades (see, e.g., [2–8]). The stabilized methods aim to relax the continuity equation so as to allow application of unstable pairs by adding extra stabilization terms. Bochev and his co-workers [4] pointed out that the unstable pairs satisfy the weaker form of the discrete inf–sup condition, and terms −  e∈Eh he∥[ph]∥2 e 1 2 and −  T∈τh hT ∥∇ ph∥T (1) reflect the inf–sup ‘deficiency’ of unstable P1/P0 and P1/P1 pairs, respectively. Stabilized methods introduce stabilization terms to counterbalance these key terms. In this paper, we propose an interesting property of the stabilized methods, that is, these key terms in (1) can be bounded by true errors. The observation provides useful arguments in a posteriori error estimates for the stabilization of low-order mixed finite element elements for the Stokes problem.

6. Numerical results

In this section, we aim to show that both the estimators ηS and ηS A work well for stabilized P1/P0 and P1/P1 finite element methods. Meanwhile, we present the practical effectivity of the average technique and show the improved estimator ηS A is more exact than the estimator ηS. We consider two numerical tests to illustrate the practical effectivity of estimators ηS and ηS A. One is a flow problem with a smooth solution. Another models a flow problem in a cracked domain with a singular solution. Moreover, to show the wide application of these estimators, we consider all the stabilized methods mentioned in Section 2. One is the penalizing jump stabilized method, namely, the stabilized method (10) with stabilization term (11). Another is the pressure gradient stabilized method, referring to the stabilized method (10) with stabilization term (12). The third one is the projection stabilized method, namely, the stabilized method (10) with stabilization term (13). The experiments are implemented by the public software Freefem++ [26]. The adaptive strategy is carried out as follows. Given a user-specified tolerance η ∗ and an initial mesh τ 0 . Refine the mesh by using the mesh refinement strategy in Freefem++ until the global error estimator η (i.e., either ηS or ηS A) satisfies η ≤ η ∗ . See [17] for details of the mesh refinement strategy.