## Weak discrete maximum principle of finite element methods in convex polyhedra

HTML articles powered by AMS MathViewer

- by
Dmitriy Leykekhman and Buyang Li
**HTML**| PDF - Math. Comp.
**90**(2021), 1-18 Request permission

## Abstract:

We prove that the Galerkin finite element solution $u_h$ of the Laplace equation in a convex polyhedron $\varOmega$, with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree $r\geqslant 1$, satisfies the following weak maximum principle: \begin{align*} \left \|u_{h}\right \|_{L^{\infty }(\varOmega )} \leqslant C\left \|u_{h}\right \|_{L^{\infty }(\partial \varOmega )} , \end{align*} with a constant $C$ independent of the mesh size $h$. By using this result, we show that the Ritz projection operator $R_h$ is stable in $L^\infty$ norm uniformly in $h$ for $r\geq 2$, i.e., \begin{align*} \|R_hu\|_{L^{\infty }(\varOmega )} \leqslant C\|u\|_{L^{\infty }(\varOmega )} . \end{align*} Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.## References

- 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, Max Winkler, and Johannes Pfefferer,
*Error estimates for the postprocessing approach applied to Neumann boundary control problems in polyhedral domains*, IMA J. Numer. Anal.**38**(2018), no. 4, 1984–2025. MR**3867389**, DOI 10.1093/imanum/drx059 - Niklas Behringer, Dmitriy Leykekhman, and Boris Vexler,
*Global and local pointwise error estimates for finite element approximations to the Stokes problem on convex polyhedra*, SIAM J. Numer. Anal.**58**(2020), no. 3, 1531–1555. MR**4101369**, DOI 10.1137/19M1274456 - Jan Brandts, Sergey Korotov, Michal Křížek, and Jakub Šolc,
*On nonobtuse simplicial partitions*, SIAM Rev.**51**(2009), no. 2, 317–335. MR**2505583**, DOI 10.1137/060669073 - 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 - Constantin Christof,
*$L^\infty$-error estimates for the obstacle problem revisited*, Calcolo**54**(2017), no. 4, 1243–1264. MR**3735814**, DOI 10.1007/s10092-017-0228-1 - Philippe G. Ciarlet,
*Discrete maximum principle for finite-difference operators*, Aequationes Math.**4**(1970), 338–352. MR**292317**, DOI 10.1007/BF01844166 - P. G. Ciarlet and P.-A. Raviart,
*Maximum principle and uniform convergence for the finite element method*, Comput. Methods Appl. Mech. Engrg.**2**(1973), 17–31. MR**375802**, DOI 10.1016/0045-7825(73)90019-4 - Monique Dauge,
*Neumann and mixed problems on curvilinear polyhedra*, Integral Equations Operator Theory**15**(1992), no. 2, 227–261. MR**1147281**, DOI 10.1007/BF01204238 - Andrei Drăgănescu, Todd F. Dupont, and L. Ridgway Scott,
*Failure of the discrete maximum principle for an elliptic finite element problem*, Math. Comp.**74**(2005), no. 249, 1–23. MR**2085400**, DOI 10.1090/S0025-5718-04-01651-5 - Alan Demlow,
*Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quadilinear elliptic problems*, SIAM J. Numer. Anal.**44**(2006), no. 2, 494–514. MR**2218957**, DOI 10.1137/040610064 - Jens Frehse and Rolf Rannacher,
*Asymptotic $L^{\infty }$-error estimates for linear finite element approximations of quasilinear boundary value problems*, SIAM J. Numer. Anal.**15**(1978), no. 2, 418–431. MR**502037**, DOI 10.1137/0715026 - P. Grisvard,
*Elliptic problems in nonsmooth domains*, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR**775683** - J. Guzmán, D. Leykekhman, J. Rossmann, and A. H. Schatz,
*Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods*, Numer. Math.**112**(2009), no. 2, 221–243. MR**2495783**, DOI 10.1007/s00211-009-0213-y - W. Höhn and H.-D. Mittelmann,
*Some remarks on the discrete maximum-principle for finite elements of higher order*, Computing**27**(1981), no. 2, 145–154 (English, with German summary). MR**632125**, DOI 10.1007/BF02243548 - David Jerison and Carlos E. Kenig,
*The inhomogeneous Dirichlet problem in Lipschitz domains*, J. Funct. Anal.**130**(1995), no. 1, 161–219. MR**1331981**, DOI 10.1006/jfan.1995.1067 - T. Kashiwabara and T. Kemmochi,
*Maximum norm error estimates for the finite element approximation of parabolic problems on smooth domains*, Preprint, 2018, arXiv:1805.01336. - Sergey Korotov and Michal Křížek,
*Acute type refinements of tetrahedral partitions of polyhedral domains*, SIAM J. Numer. Anal.**39**(2001), no. 2, 724–733. MR**1860255**, DOI 10.1137/S003614290037040X - Sergey Korotov, Michal Křížek, and Pekka Neittaanmäki,
*Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle*, Math. Comp.**70**(2001), no. 233, 107–119. MR**1803125**, DOI 10.1090/S0025-5718-00-01270-9 - Dmitriy Leykekhman and Boris Vexler,
*Finite element pointwise results on convex polyhedral domains*, SIAM J. Numer. Anal.**54**(2016), no. 2, 561–587. MR**3470741**, DOI 10.1137/15M1013912 - Dmitriy Leykekhman and Boris Vexler,
*Pointwise best approximation results for Galerkin finite element solutions of parabolic problems*, SIAM J. Numer. Anal.**54**(2016), no. 3, 1365–1384. MR**3498514**, DOI 10.1137/15M103412X - Buyang Li,
*Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra*, Math. Comp.**88**(2019), no. 315, 1–44. MR**3854049**, DOI 10.1090/mcom/3316 - Dominik Meidner and Boris Vexler,
*Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations*, ESAIM Math. Model. Numer. Anal.**52**(2018), no. 6, 2307–2325. MR**3905187**, DOI 10.1051/m2an/2018040 - Joachim A. Nitsche and Alfred H. Schatz,
*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**373325**, DOI 10.1090/S0025-5718-1974-0373325-9 - Vitoriano Ruas Santos,
*On the strong maximum principle for some piecewise linear finite element approximate problems of nonpositive type*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**29**(1982), no. 2, 473–491. MR**672072** - Alfred H. Schatz,
*A weak discrete maximum principle and stability of the finite element method in $L_{\infty }$ on plane polygonal domains. I*, Math. Comp.**34**(1980), no. 149, 77–91. MR**551291**, DOI 10.1090/S0025-5718-1980-0551291-3 - 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,
*On the quasi-optimality in $L_{\infty }$ of the $\dot H^{1}$-projection into finite element spaces*, Math. Comp.**38**(1982), no. 157, 1–22. MR**637283**, DOI 10.1090/S0025-5718-1982-0637283-6 - V. Thomée and L. B. Wahlbin,
*Stability and analyticity in maximum-norm for simplicial Lagrange finite element semidiscretizations of parabolic equations with Dirichlet boundary conditions*, Numer. Math.**87**(2000), no. 2, 373–389. MR**1804662**, DOI 10.1007/s002110000184 - Reiner Vanselow,
*About Delaunay triangulations and discrete maximum principles for the linear conforming FEM applied to the Poisson equation*, Appl. Math.**46**(2001), no. 1, 13–28. MR**1808427**, DOI 10.1023/A:1013775420323 - Junping Wang and Ran Zhang,
*Maximum principles for $P1$-conforming finite element approximations of quasi-linear second order elliptic equations*, SIAM J. Numer. Anal.**50**(2012), no. 2, 626–642. MR**2914278**, DOI 10.1137/110833737 - Jinchao Xu and Ludmil Zikatanov,
*A monotone finite element scheme for convection-diffusion equations*, Math. Comp.**68**(1999), no. 228, 1429–1446. MR**1654022**, DOI 10.1090/S0025-5718-99-01148-5

## Additional Information

**Dmitriy Leykekhman**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
- MR Author ID: 680657
- Email: dmitriy.leykekhman@uconn.edu
**Buyang Li**- Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
- MR Author ID: 910552
- Email: buyang.li@polyu.edu.hk
- Received by editor(s): September 18, 2019
- Received by editor(s) in revised form: February 29, 2020, and April 13, 2020
- Published electronically: July 27, 2020
- Additional Notes: This work was partially supported by NSF DMS-1913133 and a Hong Kong RGC grant (project no. 15300519).
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 1-18 - MSC (2010): Primary 65N12, 65N30
- DOI: https://doi.org/10.1090/mcom/3560
- MathSciNet review: 4166450