## Adaptive finite element methods in computational mechanics.(English)Zbl 0778.73071

Summary: We present a general approach to adaptivity for finite element methods and give applications to linear elasticity, nonlinear elasto-plasticity and nonlinear conservation laws, including numerical results.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74B05 Classical linear elasticity 74C99 Plastic materials, materials of stress-rate and internal-variable type 35L65 Hyperbolic conservation laws
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### References:

 [1] Eriksson, K., Adaptive finite element methods based on optimal error estimates for linear elliptic problems, (Technical Report (1987), Chalmers University of Technology) [2] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal., 28, 43-77 (1991) · Zbl 0732.65093 [3] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems II: A priori error estimates in $$L_{−8}(L_2)$$, (Technical Report (1992), Chalmers University of Technology) [4] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems III: Time steps variable in space and non-coercive problems, (Technical Report (1992), Chalmers University of Technology) [5] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems IV: Non-linear problems, (Technical Report (1992), Chalmers University of Technology) [8] Eriksson, K.; Johnson, C., An adaptive finite element method for linear elliptic problems, Math. Comp., 50, 361-383 (1988) · Zbl 0644.65080 [9] Eriksson, K.; Johnson, C., Error estimates and automatic time step control for nonlinear parabolic problems, I, SIAM J. Numer. Anal., 24, 12-23 (1987) · Zbl 0618.65104 [10] Johnson, C., Error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 25, 908-926 (1988) · Zbl 0661.65076 [11] Johnson, C., Adaptive finite element methods for diffusion and convection problems, Comput. Methods Appl. Mech. Engrg., 82, 301-322 (1990) · Zbl 0717.76078 [12] Johnson, C., Adaptive finite element methods for the obstacle problem, (Technical Report (1991), Chalmers University of Technology: Chalmers University of Technology Göteborg), to appear in $$M^3$$ AS. · Zbl 0767.65056 [13] Johnson, C.; Hansbo, P., Adaptive finite element methods for small strain elasto-plasticity, (Proc. IUTAM Conf. on Finite Inelastic Deformations — Theory and Applications (1991), Univ. of Hannover), in press. [14] Johnson, C., Discontinuous Galerkin finite element methods for second order hyperbolic problems, (Technical Report (1991), Chalmers University of Technology: Chalmers University of Technology Göteborg), to appear in Comput. Methods Appl. Mech. Engrg. · Zbl 0787.65070 [15] Hansbo, P.; Johnson, C., Adaptive streamline diffusion methods for compressible flow using conservation variables, Comput. Methods Appl. Mech. Engrg., 87, 267-280 (1991) · Zbl 0760.76046 [17] Johnson, C., A new approach to algorithms for convection problems which are based on exact transport + projection, Comput. Methods Appl. Mech. Engrg., 100, 45-62 (1992) · Zbl 0825.76413 [18] Babuška, I.; Rheinboldt, W. C., A posteriori error estimators for the finite element method, Internat. J. Numer. Methods Engrg., 12, 1597-1615 (1978) · Zbl 0396.65068 [19] Babuška, I., Feedback, adaptivity and a posteriori estimates in finite elements: Aims, theory, and experience, (Babuška, I.; etal., Accuracy Estimates and Adaptive Refinements in Finite Element Computations (1986), Wiley: Wiley New York), 3-23 [20] Bank, R.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, 283-301 (1985) · Zbl 0569.65079 [21] Oden, J. T.; Demkowicz, L.; Rachowicz, W.; Westermann, T. A., Toward a universal $$h-p$$ adaptive finite element strategy, Part 2. A posteriori error estimation, Comput. Methods Appl. Mech. Engrg., 77, 113-180 (1989) · Zbl 0723.73075 [22] Zienkiewicz, O. C.; Zhu, J. Z., A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg., 24, 337-357 (1987) · Zbl 0602.73063 [23] Ainsworth, M.; Zhu, J. Z.; Craig, A. W.; Zienkiewicz, O. C., Analysis of the Zienkiewicz-Zhu a posteriori error estimator in the finite element method, Internat. J. Numer. Methods Engrg., 28, 2161-2174 (1989) · Zbl 0716.73082 [24] Verfürth, R., A posteriori error estimators for the Stokes equation, Numer. Math., 55, 309-325 (1989) · Zbl 0674.65092 [25] Rank, E.; Zienkiewicz, O. C., A simple error estimator in the finite element method, Comm. Appl. Numer. Methods, 3, 243-249 (1987) · Zbl 0623.65120 [26] Peraire, J.; Vahdati, M.; Morgan, K.; Zienkiewicz, O. C., Adaptive remeshing for compressible flow computations, J. Comput. Phys., 72, 449-466 (1987) · Zbl 0631.76085 [27] Simo, J. C., A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II: Computational aspects, Comput. Methods Appl. Mech. Engrg., 68, 1-31 (1988) · Zbl 0644.73043 [28] Temam, R., Mathematical problems in plasticity (1985), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0457.73017 [29] Jespersen, D., Ritz-Galerkin methods for singular boundary value problems, SIAM J. Numer. Anal., 15, 813-834 (1978) · Zbl 0393.65043
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