The patch test for mixed formulations. (English) Zbl 0614.65115
A simple extension of patch test concepts to mixed elements is shown to provide a necessary and sufficient test for convergence. Many results of stability failure available previously from rather complex mathematics came out clearly in a conceptual patch test as failing to satisfy a necessary and simple condition. The general algebraic conditions of Babuska and Brezzi are given a simple form and an application of the patch tests serves to point out the instability of several well known formulations for incompressible problems.
Reviewer: N.F.F.Ebecken
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 35J25 | Boundary value problems for second-order elliptic equations |
References:
| [1] | Taylor, Int. j. numer. methods eng. 22 pp 39– (1986) |
| [2] | I, Numer. math. 20 pp 179– (1973) |
| [3] | I, Num. Math. 16 pp 322– (1971) |
| [4] | F, RAIRD 8 pp 129– (1974) |
| [5] | and , ’Mixed and irreducible formulations in finite element analysis’, in and (eds), Hybrid and Mixed Finite Element Methods Wiley, 1983, Chapter 21, pp. 405-431. |
| [6] | Zienkiewicz, Comp. Meth. Appl. Mech. Eng. 51 pp 3– (1985) |
| [7] | and , ’Mixed finite element solution of fluid flow problems’, in , and (eds), Finite Elements in Fluids, Vol. 4, Wiley, 1982, chapter 1, pp. 1-20. |
| [8] | Zienkiewicz, Am. Soc. Mech. Eng. A.M.D. 51 pp 157– (1982) |
| [9] | and , ’The convergence and stability of elements in mixed finite element analysis’, Report, Inst. Num. Meth. in Eng. C/R/511/85, Swansea, 1985. |
| [10] | ’R.I.P. methods for Stokesian flow’, in , and (eds), Finite Elements in Fluids, Vol. 4, Wiley, 1982, Chapter 15, pp. 305-318. |
| [11] | and , ’Newer and newer elements for incompressible flow’, in , and (eds), Finite Elements in Fluids, Vol. 6, Wiley, 1985, chapter 7, pp. 171-188. |
| [12] | Malkus, Comp. Meth. Appl. Mech. Eng. 15 (1978) |
| [13] | Zienkiewicz, Comp. Struct. 19 pp 303– (1984) |
| [14] | Taylor, Computers in Fluids 1 pp 73– (1973) |
| [15] | ’Displacement and equilibrium models in finite element method’ in and (eds), Stress Analysis, Wiley, 1986, chapter 9, pp. 145-197. |
| [16] | The Finite Element Method, 3rd edn, McGraw Hill, 1977. |
| [17] | Hinton, Int. j. numer. methods eng. 8 pp 461– (1974) |
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