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An adaptive stabilized conforming finite element method via residual minimization on dual discontinuous Galerkin norms. (English) Zbl 1436.65173
Summary: We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has $$\inf$$-$$\sup$$ stability. We formulate this residual minimization as a stable saddle-point problem, which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement. Numerical results on an advection-reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on uniformly refined meshes and smooth solutions. Moreover, the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers.
##### 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 76M10 Finite element methods applied to problems in fluid mechanics
CHOLMOD; FEniCS
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##### References:
 [1] Ern, A.; Guermond, J.-L., Theory and Practice of Finite Elements, Vol. 159 (2004), Springer Science [2] Hughes, T. J.R.; Scovazzi, G.; Franca, L. P., Multiscale and stabilized methods, (Encyclopedia of Computational Mechanics Second Edition (2017), American Cancer Society), 1-64 [3] Ern, A.; Guermond, J.-L., Linear stabilization for first-order PDEs, (Handbook of Numerical Methods for Hyperbolic Problems. Handbook of Numerical Methods for Hyperbolic Problems, Handb. Numer. Anal., vol. 17 (2016), Elsevier/North-Holland: Elsevier/North-Holland Amsterdam), 265-288 [4] Džiškariani, A., The least square and Bubnov-Galerkin methods, Ž. Vyčisl. Mat. Mat. Fiz., 8, 1110-1116 (1968) · Zbl 0182.21202 [5] Lučka, A., The rate of convergence to zero of the residual and the error for the Bubnov-Galerkin method and the method of least squares, (Proc. Sem. Differential and Integral Equations, No. I (Russian) (1969), Akad. Nauk Ukrain. SSR Inst. Mat.: Akad. Nauk Ukrain. SSR Inst. Mat. Kiev, Ukraine), 113-122 [6] Bramble, J. H.; Schatz, A. H., Rayleigh-Ritz-Galerkin-methods for Dirichlet’s problem using subspaces without boundary conditions, Comm. Pure Appl. Math., 23, 653-675 (1970) · Zbl 0204.11102 [7] Jiang, B., The Least-Squares Finite Element Method (1998), Springer · Zbl 0904.76003 [8] Hughes, T. J.R.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/Least-Squares method for advection-diffusive equations, Comput. Methods Appl. Mech. Engrg., 73, 173-189 (1989) · Zbl 0697.76100 [9] Cohen, A.; Dahmen, W.; Welper, G., Adaptivity and variational stabilization for convection-diffusion equations, M2AN Math. Model. Numer. Anal., 46, 5, 1247-1273 (2012) · Zbl 1270.65065 [10] Chan, J.; Evans, J. A.; Qiu, W., A dual Petrov-Galerkin finite element method for the convection-diffusion equation, Comput. Math. Appl., 68, 11, 1513-1529 (2014) · Zbl 1364.65192 [11] Zitelli, J.; Muga, I.; Demkowicz, L.; Gopalakrishnan, J.; Pardo, D.; Calo, V. M., A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D, J. Comput. Phys., 230, 7, 2406-2432 (2011) · Zbl 1316.76054 [12] Demkowicz, L.; Heuer, N., Robust DPG method for convection-dominated diffusion problems, SIAM J. Numer. Anal., 51, 5, 2514-2537 (2013) · Zbl 1290.65088 [13] Chan, J.; Heuer, N.; Bui-Thanh, T.; Demkowicz, L., A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms, Comput. Math. Appl., 67, 4, 771-795 (2014) · Zbl 1350.65120 [14] Demkowicz, L.; Gopalakrishnan, J., An overview of the discontinuous Petrov Galerkin method, (Feng, X.; Karakashian, O.; Xing, Y., Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations: 2012 John H Barrett Memorial Lectures. Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations: 2012 John H Barrett Memorial Lectures, The IMA Volumes in Mathematics and its Applications, vol. 157 (2014), Springer: Springer Cham), 149-180 · Zbl 1282.65152 [15] Guermond, J.-L., Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2AN Math. Model. Numer. Anal., 33, 6, 1293-1316 (1999) · Zbl 0946.65112 [16] Becker, R.; Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 38, 4, 173-199 (2001) · Zbl 1008.76036 [17] Matthies, G.; Skrzypacz, P.; Tobiska, L., A unified convergence analysis for local projection stabilisations applied to the Oseen problem, M2AN Math. Model. Numer. Anal., 41, 4, 713-742 (2007) · Zbl 1188.76226 [18] Burman, E., A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty, SIAM J. Numer. Anal., 43, 5, 2012-2033 (2005), (electronic) · Zbl 1111.65102 [19] Burman, E.; Hansbo, P., Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems, Comput. Methods Appl. Mech. Engrg., 193, 15-16, 1437-1453 (2004) · Zbl 1085.76033 [20] Reed, W. H.; Hill, T. R., Triangular Mesh Methods for the Neutron Transport EquationTechnical Report LA-UR-73-0479 (1973), Los Alamos Scientific Laboratory: Los Alamos Scientific Laboratory Los Alamos, NM, http://lib-www.lanl.gov/cgi-bin/getfile?00354107.pdf [21] Lesaint, P.; Raviart, P.-A., On a finite element method for solving the neutron transport equation, (Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press: Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press New York), 89-123, Publication No. 33 [22] Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46, 173, 1-26 (1986) · Zbl 0618.65105 [23] Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., (Discontinuous Galerkin Methods - Theory, Computation and Applications. Discontinuous Galerkin Methods - Theory, Computation and Applications, Lecture Notes in Computer Science and Engineering, vol. 11 (2000), Springer) [24] Brezzi, F.; Marini, L. D.; Süli, E., Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14, 12, 1893-1903 (2004) · Zbl 1070.65117 [25] Ern, A.; Guermond, J.-L., Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory, SIAM J. Numer. Anal., 44, 2, 753-778 (2006) · Zbl 1122.65111 [26] Di Pietro, D. A.; Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods, Vol. 69 (2012), Springer Science [27] Demkowicz, L.; Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. Part I. The transport equation, Comput. Methods Appl. Mech. Engrg., 199, 1558-1572 (2010) · Zbl 1231.76142 [28] Demkowicz, L.; Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions, Numer. Methods Partial Differential Equations, 27, 70-105 (2011) · Zbl 1208.65164 [29] Dahmen, W.; Huang, C.; Schwab, C.; Welper, G., Adaptive Petrov-Galerkin methods for first order transport equations, SIAM J. Numer. Anal., 50, 5, 2420-2445 (2012) · Zbl 1260.65091 [30] Broersen, D.; Dahmen, W.; Stevenson, R. P., On the stability of DPG formulations of transport equations, Math. Comp. (2017) · Zbl 1383.65102 [31] Guermond, J. L., A finite element technique for solving first-order PDEs in $$L^p$$, SIAM J. Numer. Anal., 42, 2, 714-737 (2004) · Zbl 1080.65110 [32] Muga, I.; Tyler, M. J.W.; van der Zee, K. G., The discrete-dual minimal-residual method (DDMRes) for weak advection-reaction problems in banach spaces, Comput. Methods Appl. Math., 19, 3, 557-579 (2019) · Zbl 07103561 [33] Lax, P. D.; Milgram, A. N., Parabolic equations, (Selected Papers Volume I (2005), Springer), 8-31 [34] Cantin, P., Well-posedness of the scalar and the vector advection-reaction problems in Banach graph spaces, C. R. Math. Acad. Sci. Paris, 355, 892-902 (2017) · Zbl 1379.35046 [35] Devinatz, A.; Ellis, R.; Friedman, A., The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. II, Indiana Univ. Math. J., 23, 991-1011 (1973) · Zbl 0263.35026 [36] Azérad, P.; Pousin, J., Inégalité de Poincaré courbe pour le traitement variationnel de l’équation de transport, C. R. Acad. Sci. Paris I, 322, 8, 721-727 (1996) · Zbl 0852.76073 [37] Cantin, P.; Ern, A., An edge-based scheme on polyhedral meshes for vector advection-reaction equations, ESAIM Math. Model. Numer. Anal., 51, 5, 1561-1581 (2017) · Zbl 1402.65151 [38] Ayuso, B.; Marini, L. D., Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 47, 2, 1391-1420 (2009) · Zbl 1205.65308 [39] Alnæs, M. S.; Blechta, J.; Hake, J.; Johansson, A.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M. E.; Wells, G. N., The FEniCS project version 1.5, Arch. Numer. Softw., 3, 100, 9-23 (2015) [40] Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33, 3, 1106-1124 (1996) · Zbl 0854.65090 [41] Bank, R. E.; Sherman, A. H.; Weiser, A., Some refinement algorithms and data structures for regular local mesh refinement, Sci. Comput. Appl. Math. Comput. Phys. Sci., 1, 3-17 (1983) [42] Bank, R. E.; Welfert, B. D.; Yserentant, H., A class of iterative methods for solving saddle point problems, Numer. Math., 56, 7, 645-666 (1989) · Zbl 0684.65031 [43] Chen, Y.; Davis, T. A.; Hager, W. W.; Rajamanickam, S., Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate, ACM Trans. Math. Softw. (TOMS), 35, 3, 22 (2008) [44] Karakashian, O. A.; Pascal, F., A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal., 41, 6, 2374-2399 (2003) · Zbl 1058.65120 [45] Burman, E.; Ern, A., Continuous interior penalty $$h p$$-finite element methods for advection and advection-diffusion equations, Math. Comp., 76, 259, 1119-1140 (2007) · Zbl 1118.65118 [46] Ern, A.; Guermond, J.-L., Finite element quasi-interpolation and best approximation, M2AN Math. Model. Numer. Anal., 51, 4, 1367-1385 (2017) · Zbl 1378.65041
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