Minimal residual methods in negative or fractional Sobolev norms

Monsuur, Harald; Stevenson, Rob; Storn, Johannes

doi:10.1090/mcom/3904

Minimal residual methods in negative or fractional Sobolev norms
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by Harald Monsuur, Rob Stevenson and Johannes Storn

Math. Comp. 93 (2024), 1027-1052

DOI: https://doi.org/10.1090/mcom/3904

Published electronically: October 12, 2023

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Abstract:

For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.

References

Douglas N. Arnold and Richard S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), no. 6, 1276–1290. MR 1025088, DOI 10.1137/0726074
Douglas N. Arnold, Richard S. Falk, and R. Winther, Preconditioning in $H(\textrm {div})$ and applications, Math. Comp. 66 (1997), no. 219, 957–984. MR 1401938, DOI 10.1090/S0025-5718-97-00826-0
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
Pavel B. Bochev and Max D. Gunzburger, Least-squares finite element methods, Applied Mathematical Sciences, vol. 166, Springer, New York, 2009. MR 2490235, DOI 10.1007/b13382
James H. Bramble, Raytcho D. Lazarov, and Joseph E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order systems, Math. Comp. 66 (1997), no. 219, 935–955. MR 1415797, DOI 10.1090/S0025-5718-97-00848-X
James H. Bramble, Raytcho D. Lazarov, and Joseph E. Pasciak, Least-squares for second-order elliptic problems, Comput. Methods Appl. Mech. Engrg. 152 (1998), no. 1-2, 195–210. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997). MR 1602783, DOI 10.1016/S0045-7825(97)00189-8
Dirk Broersen and Rob Stevenson, A robust Petrov-Galerkin discretisation of convection-diffusion equations, Comput. Math. Appl. 68 (2014), no. 11, 1605–1618. MR 3279496, DOI 10.1016/j.camwa.2014.06.019
Carsten Carstensen, Leszek Demkowicz, and Jay Gopalakrishnan, A posteriori error control for DPG methods, SIAM J. Numer. Anal. 52 (2014), no. 3, 1335–1353. MR 3215064, DOI 10.1137/130924913
Carsten Carstensen, Dietmar Gallistl, and Mira Schedensack, Quasi-optimal adaptive pseudostress approximation of the Stokes equations, SIAM J. Numer. Anal. 51 (2013), no. 3, 1715–1734. MR 3066804, DOI 10.1137/110852346
Carsten Carstensen and Sophie Puttkammer, How to prove the discrete reliability for nonconforming finite element methods, J. Comput. Math. 38 (2020), no. 1, 142–175. MR 4076470, DOI 10.4208/jcm.1908-m2018-0174
L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions, Numer. Methods Partial Differential Equations 27 (2011), no. 1, 70–105. MR 2743600, DOI 10.1002/num.20640
L. Demkowicz, J. Gopalakrishnan, I. Muga, and J. Zitelli, Wavenumber explicit analysis of a DPG method for the multidimensional Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 213/216 (2012), 126–138. MR 2880509, DOI 10.1016/j.cma.2011.11.024
Lars Diening, Johannes Storn, and Tabea Tscherpel, Interpolation operator on negative Sobolev spaces, Math. Comp. 92 (2023), no. 342, 1511–1541. MR 4570332, DOI 10.1090/mcom/3824
Alexandre Ern, Thirupathi Gudi, Iain Smears, and Martin Vohralík, Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal $hp$ approximation estimates in $\mathbf H(\textrm {div})$, IMA J. Numer. Anal. 42 (2022), no. 2, 1023–1049. MR 4410735, DOI 10.1093/imanum/draa103
Thomas Führer, Multilevel decompositions and norms for negative order Sobolev spaces, Math. Comp. 91 (2021), no. 333, 183–218. MR 4350537, DOI 10.1090/mcom/3674
Th. F\uumlauthrer, On a mixed FEM and a FOSLS with $H^{-1}$ loads, 2022, https://arxiv.org/abs/2210.14063.
Thomas Führer, Norbert Heuer, and Michael Karkulik, MINRES for second-order PDEs with singular data, SIAM J. Numer. Anal. 60 (2022), no. 3, 1111–1135. MR 4425909, DOI 10.1137/21M1457023
J. Gopalakrishnan and W. Qiu, An analysis of the practical DPG method, Math. Comp. 83 (2014), no. 286, 537–552. MR 3143683, DOI 10.1090/S0025-5718-2013-02721-4
Gregor Gantner and Rob Stevenson, Further results on a space-time FOSLS formulation of parabolic PDEs, ESAIM Math. Model. Numer. Anal. 55 (2021), no. 1, 283–299. MR 4216839, DOI 10.1051/m2an/2020084
R. Hiptmair, Operator preconditioning, Comput. Math. Appl. 52 (2006), no. 5, 699–706. MR 2275559, DOI 10.1016/j.camwa.2006.10.008
Ralf Hiptmair and Jinchao Xu, Nodal auxiliary space preconditioning in $\textbf {H}(\textbf {curl})$ and $\textbf {H}(\textrm {div})$ spaces, SIAM J. Numer. Anal. 45 (2007), no. 6, 2483–2509. MR 2361899, DOI 10.1137/060660588
Harald Monsuur and Rob Stevenson, A pollution-free ultra-weak FOSLS discretization of the Helmholtz equation, Comput. Math. Appl. 148 (2023), 241–255. MR 4635538, DOI 10.1016/j.camwa.2023.08.013
Gerhard Starke, Multilevel boundary functionals for least-squares mixed finite element methods, SIAM J. Numer. Anal. 36 (1999), no. 4, 1065–1077. MR 1694629, DOI 10.1137/S0036142997329803
Rob P. Stevenson, First-order system least squares with inhomogeneous boundary conditions, IMA J. Numer. Anal. 34 (2014), no. 3, 863–878. MR 3232437, DOI 10.1093/imanum/drt042
Rob Stevenson and Raymond van Venetië, Uniform preconditioners for problems of negative order, Math. Comp. 89 (2020), no. 322, 645–674. MR 4044445, DOI 10.1090/mcom/3481
Rob Stevenson and Raymond van Venetië, Uniform preconditioners for problems of positive order, Comput. Math. Appl. 79 (2020), no. 12, 3516–3530. MR 4094780, DOI 10.1016/j.camwa.2020.02.009
Rob Stevenson and Jan Westerdiep, Minimal residual space-time discretizations of parabolic equations: asymmetric spatial operators, Comput. Math. Appl. 101 (2021), 107–118. MR 4323079, DOI 10.1016/j.camwa.2021.09.014
Rob Stevenson and Jan Westerdiep, Stability of Galerkin discretizations of a mixed space-time variational formulation of parabolic evolution equations, IMA J. Numer. Anal. 41 (2021), no. 1, 28–47. MR 4205051, DOI 10.1093/imanum/drz069
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
F. Tantardini. Quasi-optimality in the backward Euler-Galerkin method for linear parabolic problems. PhD thesis, Universita degli Studi di Milano, 2013.
Francesca Tantardini and Andreas Veeser, The $L^2$-projection and quasi-optimality of Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 54 (2016), no. 1, 317–340. MR 3457701, DOI 10.1137/140996811