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Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. (English) Zbl 0795.65074
Summary: Adaptive finite element methods for stationary convection-diffusion problems are designed and analyzed. The underlying discretization scheme is the shock-capturing streamline diffusion method. The adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed. A priori error estimates are used to show that the algorithms are efficient in a certain sense.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65Y20 Complexity and performance of numerical algorithms 35K20 Initial-boundary value problems for second-order parabolic equations
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##### References:
 [1] Kenneth Eriksson and Claes Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp. 50 (1988), no. 182, 361 – 383. · Zbl 0644.65080 [2] Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43 – 77. · Zbl 0732.65093 · doi:10.1137/0728003 · doi.org [3] -, Adaptive finite element methods for parabolic problems II: A priori error estimates in $${L_\infty }({L_2})$$ and $${L_\infty }({L_\infty })$$, Department of Mathematics, Chalmers University of Technology, Göteborg, 1992. [4] -, Adaptive streamline diffusion finite element methods for time dependent convection diffusion problems, (to appear). [5] P. Hansbo, Adaptivity and streamline diffusion procedures in the finite element method, Thesis, Department of Structural Mechanics, Chalmers University of Technology, Göteborg, 1989. [6] Peter Hansbo and Anders Szepessy, A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 84 (1990), no. 2, 175 – 192. · Zbl 0716.76048 · doi:10.1016/0045-7825(90)90116-4 · doi.org [7] T. Hughes et al., A new finite element method formulation for computational fluid dynamics I-IV, Comput. Methods Appl. Mech. Engrg. 54 (1986), 223-234, 341-355; 58 (1986), 305-328, 329-336. · Zbl 0572.76068 [8] Claes Johnson, Uno Nävert, and Juhani Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), no. 1-3, 285 – 312. · Zbl 0526.76087 · doi:10.1016/0045-7825(84)90158-0 · doi.org [9] Claes Johnson and Jukka Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp. 47 (1986), no. 175, 1 – 18. · Zbl 0609.76020 [10] C. Johnson, A. H. Schatz, and L. B. Wahlbin, Crosswind smear and pointwise errors in streamline diffusion finite element methods, Math. Comp. 49 (1987), no. 179, 25 – 38. · Zbl 0629.65111 [11] Claes Johnson and Anders Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987), no. 180, 427 – 444. · Zbl 0634.65075 [12] Claes Johnson, Anders Szepessy, and Peter Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp. 54 (1990), no. 189, 107 – 129. · Zbl 0685.65086 [13] Claes Johnson, Adaptive finite element methods for diffusion and convection problems, Comput. Methods Appl. Mech. Engrg. 82 (1990), no. 1-3, 301 – 322. Reliability in computational mechanics (Austin, TX, 1989). · Zbl 0717.76078 · doi:10.1016/0045-7825(90)90169-M · doi.org [14] Peter Hansbo and Claes Johnson, Adaptive streamline diffusion methods for compressible flow using conservation variables, Comput. Methods Appl. Mech. Engrg. 87 (1991), no. 2-3, 267 – 280. · Zbl 0760.76046 · doi:10.1016/0045-7825(91)90008-T · doi.org [15] R. Löhner, K. Morgan, M. Vahdati, J. P. Boris, and D. L. Book, FEM-FCT: combining unstructured grids with high resolution, Comm. Appl. Numer. Methods 4 (1988), no. 6, 717 – 729. · Zbl 0659.65085 · doi:10.1002/cnm.1630040605 · doi.org [16] R. Löhner, K. Morgan, and O. C. Zienkiewicz, Adaptive grid refinement for the compressible Euler equations, Accuracy estimates and adaptive refinements in finite element computations (Lisbon, 1984) Wiley Ser. Numer. Methods Engrg., Wiley, Chichester, 1986, pp. 281 – 297. [17] T. Strouboulis and T. Oden, A posteriori estimation of the error in finite-element approximations of convection dominated problems, Finite Element Analysis in Fluids , Univ. of Alabama in Huntsville Press, 1989, pp. 125-136. [18] A. Szepessy, Convergence of the streamline diffusion finite element method for conservation laws, Thesis, Mathematics Department, Chalmers University of Technology, Göteborg, 1989. · Zbl 0679.65072 [19] Anders Szepessy, Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. Comp. 53 (1989), no. 188, 527 – 545. · Zbl 0679.65072
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