## Finite element error estimates on the boundary with application to optimal control

HTML articles powered by AMS MathViewer

- by Thomas Apel, Johannes Pfefferer and Arnd Rösch PDF
- Math. Comp.
**84**(2015), 33-70 Request permission

## Abstract:

This paper is concerned with the discretization of linear elliptic partial differential equations with Neumann boundary condition in polygonal domains. The focus is on the derivation of error estimates in the $L^2$-norm on the boundary for linear finite elements. Whereas common techniques yield only suboptimal results, a new approach in this context is presented which allows for quasi-optimal ones, i.e., for domains with interior angles smaller than $2\pi /3$ a convergence order two (up to a logarithmic factor) can be achieved using quasi-uniform meshes. In the presence of internal angles greater than $2\pi /3$ which reduce the convergence rates on quasi-uniform meshes, graded meshes are used to maintain the quasi-optimal error bounds.

This result is applied to linear-quadratic Neumann boundary control problems with pointwise inequality constraints on the control. The approximations of the control are piecewise constant. The state and the adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of the continuous optimal control are constructed which possess superconvergence properties. Based on the improved error estimates on the boundary and optimal regularity in weighted Sobolev spaces almost second order convergence is proven for the approximations of the continuous optimal control problem. Mesh grading techniques are again used for domains with interior angles greater than $2\pi /3$. A certain regularity of the active set is assumed.

## References

- Thomas Apel, Johannes Pfefferer, and Arnd Rösch,
*Finite element error estimates for Neumann boundary control problems on graded meshes*, Comput. Optim. Appl.**52**(2012), no. 1, 3–28. MR**2925763**, DOI 10.1007/s10589-011-9427-x - Thomas Apel, Arnd Rösch, and Dieter Sirch,
*$L^\infty$-error estimates on graded meshes with application to optimal control*, SIAM J. Control Optim.**48**(2009), no. 3, 1771–1796. MR**2516188**, DOI 10.1137/080731724 - Thomas Apel, Arnd Rösch, and Gunter Winkler,
*Optimal control in non-convex domains: a priori discretization error estimates*, Calcolo**44**(2007), no. 3, 137–158. MR**2352719**, DOI 10.1007/s10092-007-0133-0 - Thomas Apel and Dieter Sirch,
*$L^2$-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes*, Appl. Math.**56**(2011), no. 2, 177–206. MR**2810243**, DOI 10.1007/s10492-011-0002-7 - Susanne C. Brenner and L. Ridgway Scott,
*The mathematical theory of finite element methods*, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR**2373954**, DOI 10.1007/978-0-387-75934-0 - Eduardo Casas and Mariano Mateos,
*Error estimates for the numerical approximation of Neumann control problems*, Comput. Optim. Appl.**39**(2008), no. 3, 265–295. MR**2396868**, DOI 10.1007/s10589-007-9056-6 - Eduardo Casas, Mariano Mateos, and Fredi Tröltzsch,
*Error estimates for the numerical approximation of boundary semilinear elliptic control problems*, Comput. Optim. Appl.**31**(2005), no. 2, 193–219. MR**2150243**, DOI 10.1007/s10589-005-2180-2 - Monique Dauge,
*Elliptic boundary value problems on corner domains*, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR**961439**, DOI 10.1007/BFb0086682 - Alan Demlow, Johnny Guzmán, and Alfred H. Schatz,
*Local energy estimates for the finite element method on sharply varying grids*, Math. Comp.**80**(2011), no. 273, 1–9. MR**2728969**, DOI 10.1090/S0025-5718-2010-02353-1 - J. Deny and J. L. Lions,
*Les espaces du type de Beppo Levi*, Ann. Inst. Fourier (Grenoble)**21**(1945), 305–370 (1955) (French). MR**74787**, DOI 10.5802/aif.55 - P. Grisvard.
*Elliptic Problems in Nonsmooth Domains*. Pitman, Boston, 1985. - M. Hinze,
*A variational discretization concept in control constrained optimization: the linear-quadratic case*, Comput. Optim. Appl.**30**(2005), no. 1, 45–61. MR**2122182**, DOI 10.1007/s10589-005-4559-5 - Michael Hinze and Ulrich Matthes,
*A note on variational discretization of elliptic Neumann boundary control*, Control Cybernet.**38**(2009), no. 3, 577–591. MR**2650352** - M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich,
*Optimization with PDE constraints*, Mathematical Modelling: Theory and Applications, vol. 23, Springer, New York, 2009. MR**2516528**, DOI 10.1007/978-1-4020-8839-1_{3} - David S. Jerison and Carlos E. Kenig,
*The Neumann problem on Lipschitz domains*, Bull. Amer. Math. Soc. (N.S.)**4**(1981), no. 2, 203–207. MR**598688**, DOI 10.1090/S0273-0979-1981-14884-9 - V. A. Kozlov, V. G. Maz′ya, and J. Rossmann,
*Elliptic boundary value problems in domains with point singularities*, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997. MR**1469972**, DOI 10.1090/surv/052 - V. A. Kozlov, V. G. Maz′ya, and J. Rossmann,
*Spectral problems associated with corner singularities of solutions to elliptic equations*, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001. MR**1788991**, DOI 10.1090/surv/085 - Alois Kufner and Anna-Margarete Sändig,
*Some applications of weighted Sobolev spaces*, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 100, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987. With German, French and Russian summaries. MR**926688** - Hengguang Li, Anna Mazzucato, and Victor Nistor,
*Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains*, Electron. Trans. Numer. Anal.**37**(2010), 41–69. MR**2777235** - Mariano Mateos and Arnd Rösch,
*On saturation effects in the Neumann boundary control of elliptic optimal control problems*, Comput. Optim. Appl.**49**(2011), no. 2, 359–378. MR**2795722**, DOI 10.1007/s10589-009-9299-5 - Vladimir Maz’ya and Jürgen Rossmann,
*Elliptic equations in polyhedral domains*, Mathematical Surveys and Monographs, vol. 162, American Mathematical Society, Providence, RI, 2010. MR**2641539**, DOI 10.1090/surv/162 - V. G. Maz’ya and B. A. Plamenevsky,
*Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points*. American Mathematical Society Translations, 123(2):89–107, 1984. - J. M. Melenk and B. Wohlmuth,
*Quasi-optimal approximation of surface based Lagrange multipliers in finite element methods*, SIAM J. Numer. Anal.**50**(2012), no. 4, 2064–2087. MR**3022210**, DOI 10.1137/110832999 - C. Meyer and A. Rösch,
*Superconvergence properties of optimal control problems*, SIAM J. Control Optim.**43**(2004), no. 3, 970–985. MR**2114385**, DOI 10.1137/S0363012903431608 - Sergey A. Nazarov and Boris A. Plamenevsky,
*Elliptic problems in domains with piecewise smooth boundaries*, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR**1283387**, DOI 10.1515/9783110848915.525 - J. Pfefferer and K. Krumbiegel,
*Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations*, arXiv:1311:6282 [math.NA]. - J. Roßmann.
*Gewichtete Sobolev-Slobodetskij-Räume und Anwendungen auf elliptische Randwertprobleme in Gebieten mit Kanten*. Habilitationsschrift, Universität Rostock, 1988. - A. H. Schatz and L. B. Wahlbin,
*Interior maximum norm estimates for finite element methods*, Math. Comp.**31**(1977), no. 138, 414–442. MR**431753**, DOI 10.1090/S0025-5718-1977-0431753-X - A. H. Schatz and L. B. Wahlbin,
*Maximum norm estimates in the finite element method on plane polygonal domains. I*, Math. Comp.**32**(1978), no. 141, 73–109. MR**502065**, DOI 10.1090/S0025-5718-1978-0502065-1 - A. H. Schatz and L. B. Wahlbin,
*Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements*, Math. Comp.**33**(1979), no. 146, 465–492. MR**502067**, DOI 10.1090/S0025-5718-1979-0502067-6 - Fredi Tröltzsch,
*Optimal control of partial differential equations*, Graduate Studies in Mathematics, vol. 112, American Mathematical Society, Providence, RI, 2010. Theory, methods and applications; Translated from the 2005 German original by Jürgen Sprekels. MR**2583281**, DOI 10.1090/gsm/112 - Lars B. Wahlbin,
*Local behavior in finite element methods*, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 353–522. MR**1115238**

## Additional Information

**Thomas Apel**- Affiliation: Universität der Bundeswehr München, 85577 Neubiberg, Germany
- Email: Thomas.Apel@unibw.de
**Johannes Pfefferer**- Affiliation: Universität der Bundeswehr München, 85577 Neubiberg, Germany
- Email: Johannes.Pfefferer@unibw.de
**Arnd Rösch**- Affiliation: Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, 45127 Essen, Germany
- Email: Arnd.Roesch@uni-due.de
- Received by editor(s): November 19, 2012
- Received by editor(s) in revised form: May 16, 2013
- Published electronically: June 27, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp.
**84**(2015), 33-70 - MSC (2010): Primary 65N30, 49M25, 65N50, 65N15
- DOI: https://doi.org/10.1090/S0025-5718-2014-02862-7
- MathSciNet review: 3266952