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Goal-oriented adaptive finite element methods for elliptic problems revisited. (English) Zbl 1316.65098
Summary: A goal-oriented a posteriori error estimation of an output functional for elliptic problems is presented. Continuous finite element approximations are used in quadrilateral and triangular meshes. The algorithm is similar to the classical dual-weighted error estimation, however the dual weight contains solutions of the proposed patch problems. The patch problems are introduced to apply Clément and Scott-Zhang type interpolation operators to estimate point values with the finite element polynomials. The algorithm is shown to be reliable, efficient and convergent.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N15 Error bounds 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
##### Software:
deal.ii; DOLFIN; Unicorn
Full Text:
##### References:
 [1] Eriksson, Kenneth; Estep, Don; Hansbo, Peter; Johnson, Claes, Introduction to adaptive methods for differential equations, Acta Numer., 4, 105-158, (1995) · Zbl 0829.65122 [2] Johnson, C.; Szepessy, A., Adaptive finite element methods for conservation laws based on a posteriori error estimates, Comm. Pure Appl. Math., 48, 199-234, (1995) · Zbl 0826.65088 [3] Becker, Roland; Rannacher, Rolf, A feed-back approach to error control in adaptive finite element methods: basic analysis and examples, East-West J. Numer. Math., 4, 237-264, (1996) · Zbl 0868.65076 [4] Giles, M.; Süli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer., 11, 145-236, (2002) · Zbl 1105.65350 [5] Oden, J. T.; Prudhomme, S., On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput. Methods Appl. Mech. Engrg., 111, 185-202, (1999) · Zbl 0945.65123 [6] Jansson, Niclas; Hoffman, Johan; Nazarov, Murtazo, Adaptive simulation of turbulent flow past a full car model, (State of the Practice Reports, SC’11, (2011), ACM New York, NY, USA), 20:1-20:8 [7] Nazarov, Murtazo; Hoffman, Johan, An adaptive finite element method for inviscid compressible flow, Internat. J. Numer. Methods Fluids, 64, 10-12, 1102-1128, (2010) · Zbl 1427.76139 [8] Nazarov, Murtazo; Hoffman, Johan, Residual-based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods, Internat. J. Numer. Methods Fluids, 71, 3, 339-357, (2013) [9] Moon, K.-S.; von Schwerin, E.; Szepessy, A.; Tempone, R., Convergence rates for an adaptive dual weighted residual finite element algorithm, BIT, 46, 2, 367-407, (2006) · Zbl 1101.65103 [10] Mommer, M. S.; Stevenson, R., A goal-oriented adaptive finite element method with convergence rates, SIAM J. Numer. Anal., 47, 2, 861-886, (2009) · Zbl 1195.65174 [11] Michael Holst, Sara Pollock, Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems, 2011. · Zbl 1337.65139 [12] Cascon, J.; Kreuzer, C.; Nochetto, R.; Siebert, K., Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46, 5, 2524-2550, (2008) · Zbl 1176.65122 [13] Bürg, M., A fully automatic $$h p$$-adaptive refinement strategy, (2012), Research Training Group 1294, Karlsruhe Institute of Technology (KIT), July [14] Schwab, Ch., $$p$$- and $$h p$$-finite element methods, (1998), Clarendon Press Oxford · Zbl 0910.73003 [15] Szabó, B.; Babuška, I., Finite element analysis, (1991), Wiley New York, NY [16] Becker, R.; Rannacher, R., A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math., 4, 237-264, (1996) · Zbl 0868.65076 [17] Becker, R.; Rannacher, R., Weighted a posteriori error control in FE methods, (Bock, H. G.; etal., Proc. ENUMATH-97, (1997), World Scientific Publ. Singapore), 621-637 · Zbl 0968.65083 [18] Rannacher, R., Error control in finite element computations, (Bulgak, H.; Zenger, C., Error Control and Adaptivity in Scientific Computing, NATO Science Series, (1999), Kluwer Academic Publ Dortrech), 247-278, Proc. NATO-Summer School · Zbl 0943.65123 [19] Bürg, M.; Dörfler, W., Convergence of an adaptive $$h p$$ finite element strategy in higher space-dimensions, Appl. Numer. Math., 61, 1132-1146, (2011) · Zbl 1230.65115 [20] Scott, L. R.; Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54, 483-493, (1990) · Zbl 0696.65007 [21] Melenk, J. M., $$h p$$-interpolation of nonsmooth functions and an application to $$h p$$-a posteriori error estimation, SIAM J. Numer. Anal., 43, 1, 127-155, (2005) · Zbl 1087.65108 [22] Melenk, J. M.; Wohlmuth, B. I., On residual-based a posteriori error estimation in $$h p$$-FEM, Adv. Comput. Math., 15, 311-331, (2001) · Zbl 0991.65111 [23] Bürg, M., Convergence of an automatic $$h p$$-adaptive finite element strategy for maxwell’s equations, Appl. Numer. Math., 72, 188-206, (2013) · Zbl 1304.78013 [24] Rivara, M.-C., Local modification of meshes for adaptive and/or multigrid finite-element methods, J. Comput. Appl. Math., 36, 1, 78-89, (1992) [25] Dörfler, W., A convergent adaptive algorithm for poisson’s equation, SIAM J. Numer. Anal., 33, 3, 1106-1124, (1996) · Zbl 0854.65090 [26] Dörfler, W.; Heuveline, V., Convergence of an adaptive $$h p$$ finite element strategy in one space dimension, Appl. Numer. Math., 57, 1108-1124, (2007) · Zbl 1123.65079 [27] Cormen, Th. H.; Leiserson, C. E.; Rivest, R.; Stein, C., Introduction to algorithms, (2001), MIT Press Cambridge, MA [28] Hartmann, Ralf; Houston, Paul, Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws, SIAM J. Sci. Comput., 24, 979-1004, (2002) · Zbl 1034.65081 [29] Logg, Anders; Wells, Garth N., DOLFIN: automated finite element computing, ACM Trans. Math. Software, 37, 2, 28, (2010), Art. 20 · Zbl 1364.65254 [30] Bangerth, W.; Hartmann, R.; Kanschat, G., Deal.II—a general purpose object oriented finite element library, ACM Trans. Math. Software, 33, 4, 24/1-24/27, (2007) · Zbl 1365.65248 [31] Hoffman, Johan; Jansson, Johan; Degirmenci, Cem; Jansson, Niclas; Nazarov, Murtazo, Unicorn: a unified continuum mechanics solver, (Logg, Anders; Mardal, Kent-Andre; Wells, Garth, Automated Solution of Differential Equations by the Finite Element Method, Lecture Notes in Computational Science and Engineering, vol. 84, (2012), Springer Berlin, Heidelberg), 339-361
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