Abstract
In this paper we formulate and analyze adaptive (space-time)
least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative
Dedicated to In memoriam Raytcho D. Lazarov (1943–2024)
Funding statement: This work is supported by the joint DFG/FWF Collaborative Research Centre CREATOR (DFG: Project-ID 492661287/TRR361; FWF: 10.55776/F90) at TU Darmstadt, TU Graz and JKU Linz.
References
[1] R. Andreev, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations, IMA J. Numer. Anal. 33 (2013), no. 1, 242–260. 10.1093/imanum/drs014Search in Google Scholar
[2] M. Augustin, A. Caiazzo, A. Fiebach, J. Fuhrmann, V. John, A. Linke and R. Umla, An assessment of discretizations for convection-dominated convection-diffusion equations, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 47–48, 3395–3409. 10.1016/j.cma.2011.08.012Search in Google Scholar
[3] I. Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. 10.1007/BF02165003Search in Google Scholar
[4] C. Bacuta, C. Bacuta and D. Hayes, Comparison of variational discretizations for a convection-diffusion problem, Rev. Roumaine Math. Pures Appl. 69 (2024), no. 3–4, 327–351. 10.59277/RRMPA.2024.327.351Search in Google Scholar
[5] C. Bacuta, D. Hayes and T. O’Grady, Saddle point least squares discretization for convection-diffusion, Appl. Anal. 103 (2024), no. 12, 2241–2268. 10.1080/00036811.2023.2291511Search in Google Scholar
[6] R. E. Bank, P. S. Vassilevski and L. T. Zikatanov, Arbitrary dimension convection-diffusion schemes for space-time discretizations, J. Comput. Appl. Math. 310 (2017), 19–31. 10.1016/j.cam.2016.04.029Search in Google Scholar
[7] P. B. Bochev and M. D. Gunzburger, Least-Squares Finite Element Methods, Appl. Math. Sci. 166, Springer, New York, 2009. 10.1007/b13382Search in Google Scholar
[8] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. 10.1007/978-3-642-36519-5Search in Google Scholar
[9] D. Braess, Finite Elements, Cambridge University, Cambridge, 2007. 10.1017/CBO9780511618635Search in Google Scholar
[10] J. H. Bramble, R. D. Lazarov and J. 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. 10.1090/S0025-5718-97-00848-XSearch in Google Scholar
[11] J. H. Bramble, R. D. Lazarov and J. E. Pasciak, Least-squares for second-order elliptic problems, Comput. Methods Appl. Mech. Engrg. 152 (1998), 195–210. 10.1016/S0045-7825(97)00189-8Search in Google Scholar
[12] A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), no. 1–3, 199–259. 10.1016/0045-7825(82)90071-8Search in Google Scholar
[13] E. Burman and A. Ern, Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 4, 681–707. 10.1051/m2an/2011047Search in Google Scholar
[14] Z. Cai, R. Lazarov, T. A. Manteuffel and S. F. McCormick, First-order system least squares for second-order partial differential equations. I, SIAM J. Numer. Anal. 31 (1994), no. 6, 1785–1799. 10.1137/0731091Search in Google Scholar
[15] A. Cangiani, E. H. Georgoulis and S. Metcalfe, Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems, IMA J. Numer. Anal. 34 (2014), no. 4, 1578–1597. 10.1093/imanum/drt052Search in Google Scholar
[16] C. Carstensen, A note on the quasi-best approximation constant, Comput. Methods Appl. Math. 25 (2025), no. 3, 581–585. 10.1515/cmam-2025-0025Search in Google Scholar
[17] C. Carstensen, L. Demkowicz and J. Gopalakrishnan, A posteriori error control for DPG methods, SIAM J. Numer. Anal. 52 (2014), no. 3, 1335–1353. 10.1137/130924913Search in Google Scholar
[18] A. Cohen, W. Dahmen and G. Welper, Adaptivity and variational stabilization for convection-diffusion equations, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 5, 1247–1273. 10.1051/m2an/2012003Search in Google Scholar
[19] W. Dahmen, H. Monsuur and R. Stevenson, Least squares solvers for ill-posed PDEs that are conditionally stable, ESAIM Math. Model. Numer. Anal. 57 (2023), no. 4, 2227–2255. 10.1051/m2an/2023050Search in Google Scholar
[20] L. Demkowicz and N. Heuer, Robust DPG method for convection-dominated diffusion problems, SIAM J. Numer. Anal. 51 (2013), no. 5, 2514–2537. 10.1137/120862065Search in Google Scholar
[21] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. 10.1137/0733054Search in Google Scholar
[22] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer, New York, 2004. 10.1007/978-1-4757-4355-5Search in Google Scholar
[23] L. P. Franca, G. Hauke and A. Masud, Revisiting stabilized finite element methods for the advective-diffusive equation, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 13–16, 1560–1572. 10.1016/j.cma.2005.05.028Search in Google Scholar
[24]
T. Führer,
On a mixed FEM and a FOSLS with
[25] T. Führer, N. Heuer and M. Karkulik, MINRES for second-order PDEs with singular data, SIAM J. Numer. Anal. 60 (2022), no. 3, 1111–1135. 10.1137/21M1457023Search in Google Scholar
[26] P. Gangl, M. Gobrial and O. Steinbach, A space-time finite element method for the eddy current approximation of rotating electric machines, Comput. Methods Appl. Math. 25 (2025), no. 2, 441–457. 10.1515/cmam-2024-0033Search in Google Scholar
[27] P. W. Hemker, A singularly perturbed model problem for numerical computation, J. Comput. Appl. Math. 76 (1996), no. 1–2, 277–285. 10.1016/S0377-0427(96)00113-6Search in Google Scholar
[28] A. Jha, V. John and P. Knobloch, Adaptive grids in the context of algebraic stabilizations for convection-diffusion-reaction equations, SIAM J. Sci. Comput. 45 (2023), no. 4, B564–B589. 10.1137/21M1466360Search in Google Scholar
[29] V. John, P. Knobloch and O. Pártl, A numerical assessment of finite element discretizations for convection-diffusion-reaction equations satisfying discrete maximum principles, Comput. Methods Appl. Math. 23 (2023), no. 4, 969–988. 10.1515/cmam-2022-0125Search in Google Scholar
[30]
J. Ku,
Adaptive least-squares finite element methods: Guaranteed upper bounds and convergence in
[31] R. D. Lazarov, L. Tobiska and P. S. Vassilevski, Streamline diffusion least-squares mixed finite element methods for convection-diffusion problems, East-West J. Numer. Math. 5 (1997), no. 4, 249–264. Search in Google Scholar
[32] R. D. Lazarov and P. S. Vassilevski, Least-squares streamline diffusion finite element approximations to singularly perturbed convection-diffusion problems, Technical Report 1999/2, Institute for Scientific Computation, Texas A&M University, College Station, 1999. Search in Google Scholar
[33] M. Łoś, P. Sepúlveda, M. Sikora and M. Paszyński, Solver algorithm for stabilized space-time formulation of advection-dominated diffusion problem, Comput. Math. Appl. 152 (2023), 67–80. 10.1016/j.camwa.2023.10.011Search in Google Scholar
[34] H. Monsuur, R. Stevenson and J. Storn, Minimal residual methods in negative or fractional Sobolev norms, Math. Comp. 93 (2024), no. 347, 1027–1052. 10.1090/mcom/3904Search in Google Scholar
[35] J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 16 (1962), 305–326. Search in Google Scholar
[36] D. R. Q. Pacheco and O. Steinbach, On the initial higher-order pressure convergence in equal-order finite element discretizations of the Stokes system, Comput. Math. Appl. 109 (2022), 140–145. 10.1016/j.camwa.2022.01.022Search in Google Scholar
[37] A. I. Pehlivanov, G. F. Carey and R. D. Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J. Numer. Anal. 31 (1994), no. 5, 1368–1377. 10.1137/0731071Search in Google Scholar
[38] H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Ser. Comput. Math. 24, Springer, Berlin, 2008. Search in Google Scholar
[39] A. Schafelner and P. S. Vassilevski, Numerical results for adaptive (negative norm) constrained first order system least squares formulations, Comput. Math. Appl. 95 (2021), 256–270. 10.1016/j.camwa.2020.08.025Search in Google Scholar
[40] O. Steinbach, A note on initial higher-order convergence results for boundary element methods with approximated boundary conditions, Numer. Methods Partial Differential Equations 16 (2000), no. 6, 581–588. 10.1002/1098-2426(200011)16:6<581::AID-NUM5>3.0.CO;2-QSearch in Google Scholar
[41] O. Steinbach, Space-time finite element methods for parabolic problems, Comput. Methods Appl. Math. 15 (2015), no. 4, 551–566. 10.1515/cmam-2015-0026Search in Google Scholar
[42] R. Stevenson and J. 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. 10.1093/imanum/drz069Search in Google Scholar
[43] I. Toulopoulos, SUPG space-time scheme on anisotropic meshes for general parabolic equations, J. Numer. Math. 33 (2025), no. 4, 313–336. 10.1515/jnma-2024-0053Search in Google Scholar
[44] R. Verfürth, Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal. 43 (2005), no. 4, 1766–1782. 10.1137/040604261Search in Google Scholar
[45] J. Xu and L. Zikatanov, Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003), no. 1, 195–202. 10.1007/s002110100308Search in Google Scholar
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