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Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. (English) Zbl 0608.65013
The authors study the divergence operator acting on continuous piecewise polynomials of degree \(p+1\), \(p\geq 3\), over triangulations of a plane polygonal domain. A combinatorial argument is used to verify a characterization of the range of the divergence operator. It is shown that for every general families of meshes it is possible to find a maximal right inverse for the divergence operator with a \(B(L_ 2,H^ 1)\) norm which is bounded independently of the mesh size. The norm of this right inverse grows at most algebraically with p, but it necessarily blows up as a certain measure of singularity of the meshes approaches 0.
Reviewer: P.Narain

65D25 Numerical differentiation
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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[1] D. N. ARNOLD, L. R. SCOTT, M. VOGELIUS, Regular solutions of div u = f with Dirichlet boundary conditions on a polygon, Tech. Note, University of Maryland, to appear. · Zbl 0702.35208
[2] I. BABUSKA, K. AZIZ, Survey lectures on the mathematical foundations of the finite element method. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz, editor, Academic Press, 1972. Zbl0268.65052 MR347104 · Zbl 0268.65052
[3] J. M. BOLAND, R.A. NICOLAIDES, Stability of finite elements under devergence constraints, SIAM J. Num. Anal. 20 (1983), pp. 722-731. Zbl0521.76027 MR708453 · Zbl 0521.76027
[4] [4] M. CROUZEIX, P. A. RAVIART, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, I. R.A.LR.O. Sér. Rouge 7 (1973), pp. 33-75. Zbl0302.65087 MR343661 · Zbl 0302.65087
[5] P.C. DUNNE, Reply to comments by B. Irons on his paper < Complete polynomial displacement fields for finite element method > , Aero. J. Roy. Aero. Soc.72 (1973) pp. 710-711.
[6] [6] G. J. FIX, M. D. GUNZBURGER, R. A. NICOLAIDES, On mixed finite element methods for first order elliptic systems. Numer. Math. 37 (1981), pp. 29-48. Zbl0459.65072 MR615890 · Zbl 0459.65072
[7] V. GIRAULT, P. A. RAVIART, Finite Element Approximation of the Navier-Stokes Equation. Lecture Notes in Mathematics, 749, Springer-Verlag, 1979. Zbl0413.65081 MR548867 · Zbl 0413.65081
[8] P. GRISVARD, Boundary value problems in non-smooth domains, Lecture Notes # 19, University of Maryland, 1980.
[9] B. MERCIER, A conforming finite element method for two dimensional, incompressible elasticity, Int. J. Num Meths. Eng. 14 (1979), pp. 942-945. Zbl0397.73065 MR533310 · Zbl 0397.73065
[10] J. MORGAN R. SCOTT, A nodal basis for C 1 piecewise polynomials of degree n \geq 5 no 5. Math. Comput. 29 (1975), pp. 736-740. Zbl0307.65074 MR375740 · Zbl 0307.65074
[11] J. MORGAN R. SCOTT, The dimension of the space of C 1 piecewise polynomials (Preprint).
[12] L. R. SCOTT, M. VOGELIUS, Conforming finite element methods for incompressible and nearly incompressible continua. Proceedings of the 1983 Summer Seminar on Large-scale Computations in Fluid Mechanics, S. Osher, editor, Lect. Appl. Math. 22, to appear. Zbl0582.76028 MR818790 · Zbl 0582.76028
[13] E. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. Zbl0207.13501 MR290095 · Zbl 0207.13501
[14] R. STENBERG, Analysis of mixed finite element methods for the Stokes problem : A unified approach. To appear, Math. Comp. Zbl0535.76037 MR725982 · Zbl 0535.76037
[15] G. STRANG, Piecewise polynomials and the finite element method, Bull. AMS 79 (1973), pp, 1128-1137. Zbl0285.41009 MR327060 · Zbl 0285.41009
[16] B. A. SZABO, P. K. BASU, D. A. DUNAVANT, D. VASILOPOULOS, Adaptive finite element technology in integrated design and analysis, Report WU/CCM-81/1. Washington Univestity, St. Louis.
[17] R. TEMAM, Navier-Stokes Equations, North-Holland, 1977. Zbl0383.35057 MR769654 · Zbl 0383.35057
[18] [18] M. VOGELIUS, A right-inverse for the divergence operator in spaces of piecewise polynomials. Application to the p-version of the finite element method. Numer. Math. 41 (1983), pp. 19-37. Zbl0504.65060 MR696548 · Zbl 0504.65060
[19] [19] M. VOGELIUS, An analysis of thep-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal error estimates. Numer. Math. 41 (1983), pp. 39-53. Zbl0504.65061 MR696549 · Zbl 0504.65061
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