## Local problems on stars: A posteriori error estimators, convergence, and performance.(English)Zbl 1019.65083

The authors consider the numerical solution of a generalised two-dimensional Poisson equation with the Dirichlet boundary condition over a bounded polygonal domain by means of triangular finite elements and produce estimates for the error. Estimators with a self-equilibration property are suggested and a convergent algorithm is given for the solution of the problem. A measure of the error is defined and an indication as to how it may be reduced is given, together with rules for oscillation reduction.
Results are established for boundary stars. The paper closes with the results of some numerical experiments and it is suggested that the performance of the error estimators and quasi-optimality of the algorithm is good. The examples of the application of the method, the problem of a crack and a problem involving discontinuous coefficients are used.

### MSC:

 65N15 Error bounds for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65Y20 Complexity and performance of numerical algorithms

ALBERT
Full Text:

### References:

 [1] Mark Ainsworth and Ivo Babuška, Reliable and robust a posteriori error estimating for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal. 36 (1999), no. 2, 331 – 353. · Zbl 0948.65114 [2] Mark Ainsworth and J. Tinsley Oden, A unified approach to a posteriori error estimation using element residual methods, Numer. Math. 65 (1993), no. 1, 23 – 50. · Zbl 0797.65080 [3] I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987), no. 1, 1 – 40. · Zbl 0593.65064 [4] R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283 – 301. · Zbl 0569.65079 [5] Carsten Carstensen and Stefan A. Funken, Fully reliable localized error control in the FEM, SIAM J. Sci. Comput. 21 (1999/00), no. 4, 1465 – 1484. · Zbl 0956.65099 [6] Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106 – 1124. · Zbl 0854.65090 [7] W. Dörfler and R.H. Nochetto, Small data oscillation implies the saturation assumption, Numer. Math., 91 (2002), 1-12. · Zbl 0995.65109 [8] R.G. Durán and M.A. Muschietti, An explicit right inverse of the divergence operator in $$W_0^{1,p}(\Omega)^n$$, Studia Math., 148 (2001), 207-219. · Zbl 0985.35059 [9] R. Bruce Kellogg, On the Poisson equation with intersecting interfaces, Applicable Anal. 4 (1974/75), 101 – 129. Collection of articles dedicated to Nikolai Ivanovich Muskhelishvili. · Zbl 0307.35038 [10] Alois Kufner, Weighted Sobolev spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. With German, French and Russian summaries. · Zbl 0455.46034 [11] Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466 – 488. · Zbl 0970.65113 [12] P. Morin, R.H. Nochetto and K.G. Siebert, Basic principles for convergence of adaptive higher-order FEM – Application to linear elasticity, in preparation. [13] Jindřich Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm. Sup. Pisa (3) 16 (1962), 305 – 326 (French). · Zbl 0112.33101 [14] Ricardo H. Nochetto, Removing the saturation assumption in a posteriori error analysis, Istit. Lombardo Accad. Sci. Lett. Rend. A 127 (1993), no. 1, 67 – 82 (1994) (English, with Italian summary). · Zbl 0878.65088 [15] A. Schmidt and K.G. Siebert, ALBERT – Software for scientific computations and applications, Acta Math. Univ. Comenianae 70 (2001), 105-122. · Zbl 0993.65134 [16] A. Schmidt and K.G. Siebert, ALBERT: An adaptive hierarchical finite element toolbox, Documentation, Preprint 06/2000 Universität Freiburg, 244 p. [17] T. Strouboulis, I. Babuška, and S. K. Gangaraj, Guaranteed computable bounds for the exact error in the finite element solution. II. Bounds for the energy norm of the error in two dimensions, Internat. J. Numer. Methods Engrg. 47 (2000), no. 1-3, 427 – 475. Richard H. Gallagher Memorial Issue. , https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/33.3.CO;2-T · Zbl 0962.65069 [18] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, 1996. · Zbl 0853.65108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.