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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 82 (1990) 323-340 NORTH-HOLLAND
A POSTERIORI ERROR ESTIMATES FOR THE STOKES EQUATIONS: A COMPARISON
Randolph E. BANK* and Bruno D. WELFERT* Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, U.S.A. Received 1 November 1989
Several a posteriori error estimates for the Stokes equations have been derived by several authors. In this paper we compare some estimates based on the solution of local Stokes systems with estimates based on the residuals of the discretized finite element equations. Their performance as local indicators as well as global estimates is investigated.
I. Introduction
When numerically solving a set of partial differential equations through a finite element strategy associated with a weak formulation, one usually faces the problem of increasing the accuracy of the solution without adding unnecessary degrees of freedom in non-critical parts of the computational domain. In order to identify these regions, indicators were created which allow their automatic determination by computing some function of the characteristic features of the solution, such as indicators based on the gradient of the Mach number in Computational Fluid Dynamics (CFD) [1, 2], indicators derived from a priori error estimates, or indicators involving residuals of the discretized equations [3, 4]. More recently the trend has been to derive a posteriori error estimates based on more mathematical criteria, by solving small local problems resembling the original global one, but involving higher order finite elements [5-8]. In this paper we compare a few of these estimates obtained for the Stokes problem. The finite element scheme used is the classical mini-element formulation, which is recalled in Section 2. Three estimates based on the resolution of a local Stokes problem along with one based only on residuals are presented in Section 3. In Section 4 a few comparison inequalities are stated, and Section 5 examines their numerical behavior on test problems for which an exact solution is known, and on typical examples of CFD as well. *The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440, by Avions Marcel Dassault - Breguet Aviation, 78 quai Marcel Dassault, 92214 St Cloud, France and by Direction des Recherches Etudes et Techniques, 26 boulevard Victor, 75996 Paris Armres, France. *The work of this author was supported by Avions Marcel Dassault- Breguet Aviation, 78 quai Marcel Dassault, 92214 St Cloud, France and by Direction des Recherches Etudes et Techniques, 26 boulevard Victor, 75996 Paris Armres, France. 0045-7825/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)
R.E. Bank, B.D. Welfert, A posteriori error estimates for the Stokes equations
324
2. The mini-element diseretization of the Stokes equations
We consider the classical Stokes problem: Find u (velocity field, 2 components) E (~ox(/2))2 (the usual Sobolev space) and p (pressure field)~ ~2(/2) (the usual Lebesque space) such that in /2,
-vAu+Vp=f
in /2,
V.u=O u= g
(1)
on 0/2
in a bounded d o m a i n / 2 C ~ 2 (/,, is a viscosity parameter). A weak formulation of (1) can be derived using integration by parts, and can be shown to satisfy an LBB condition, thus providing a unique solution to the resulting system [9] satisfying fa p d / 2 =0. Let 3-denote a triangulation of the domain /2, such that any two triangles in ff share at most a vertex or an edge. Let h, be the diameter of a triangle r ~ 3- and h = max,ee~ h r. E is the set of interior edges. For e E E, h e denotes the length of e. We suppose also that the triangulation 3-satisfies a minimal angle condition, i.e., the smallest angle in triangle "r ~ Sr is bounded away from zero by some constant independent of h. This implies in particular that C l h ~ ~ / /\\>~
A POSTERIORI ERROR ESTIMATES FOR THE STOKES EQUATIONS: A COMPARISON
Randolph E. BANK* and Bruno D. WELFERT* Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, U.S.A. Received 1 November 1989
Several a posteriori error estimates for the Stokes equations have been derived by several authors. In this paper we compare some estimates based on the solution of local Stokes systems with estimates based on the residuals of the discretized finite element equations. Their performance as local indicators as well as global estimates is investigated.
I. Introduction
When numerically solving a set of partial differential equations through a finite element strategy associated with a weak formulation, one usually faces the problem of increasing the accuracy of the solution without adding unnecessary degrees of freedom in non-critical parts of the computational domain. In order to identify these regions, indicators were created which allow their automatic determination by computing some function of the characteristic features of the solution, such as indicators based on the gradient of the Mach number in Computational Fluid Dynamics (CFD) [1, 2], indicators derived from a priori error estimates, or indicators involving residuals of the discretized equations [3, 4]. More recently the trend has been to derive a posteriori error estimates based on more mathematical criteria, by solving small local problems resembling the original global one, but involving higher order finite elements [5-8]. In this paper we compare a few of these estimates obtained for the Stokes problem. The finite element scheme used is the classical mini-element formulation, which is recalled in Section 2. Three estimates based on the resolution of a local Stokes problem along with one based only on residuals are presented in Section 3. In Section 4 a few comparison inequalities are stated, and Section 5 examines their numerical behavior on test problems for which an exact solution is known, and on typical examples of CFD as well. *The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440, by Avions Marcel Dassault - Breguet Aviation, 78 quai Marcel Dassault, 92214 St Cloud, France and by Direction des Recherches Etudes et Techniques, 26 boulevard Victor, 75996 Paris Armres, France. *The work of this author was supported by Avions Marcel Dassault- Breguet Aviation, 78 quai Marcel Dassault, 92214 St Cloud, France and by Direction des Recherches Etudes et Techniques, 26 boulevard Victor, 75996 Paris Armres, France. 0045-7825/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)
R.E. Bank, B.D. Welfert, A posteriori error estimates for the Stokes equations
324
2. The mini-element diseretization of the Stokes equations
We consider the classical Stokes problem: Find u (velocity field, 2 components) E (~ox(/2))2 (the usual Sobolev space) and p (pressure field)~ ~2(/2) (the usual Lebesque space) such that in /2,
-vAu+Vp=f
in /2,
V.u=O u= g
(1)
on 0/2
in a bounded d o m a i n / 2 C ~ 2 (/,, is a viscosity parameter). A weak formulation of (1) can be derived using integration by parts, and can be shown to satisfy an LBB condition, thus providing a unique solution to the resulting system [9] satisfying fa p d / 2 =0. Let 3-denote a triangulation of the domain /2, such that any two triangles in ff share at most a vertex or an edge. Let h, be the diameter of a triangle r ~ 3- and h = max,ee~ h r. E is the set of interior edges. For e E E, h e denotes the length of e. We suppose also that the triangulation 3-satisfies a minimal angle condition, i.e., the smallest angle in triangle "r ~ Sr is bounded away from zero by some constant independent of h. This implies in particular that C l h ~ ~ / /\\>~
