Error estimates for finite element approximations of parabolic equations with measure data
HTML articles powered by AMS MathViewer
- by Wei Gong PDF
- Math. Comp. 82 (2013), 69-98 Request permission
Abstract:
In this paper we study the a priori error estimates for the finite element approximations of parabolic equations with measure data, especially we consider problems with separate measure data in time and space, respectively. The solutions of these kinds of problems exhibit low regularities due to the existence of measure data, this introduces some difficulties in both theoretical and numerical analysis. For both cases we use standard piecewise linear and continuous finite elements for the space discretization and derive the a priori error estimates for the semi-discretization problems, while the backward Euler method is then used for time discretization and a priori error estimates for the fully discrete problems are also derived. Numerical results are provided at the end of the paper to confirm our theoretical findings.References
- Rodolfo Araya, Edwin Behrens, and Rodolfo Rodríguez, A posteriori error estimates for elliptic problems with Dirac delta source terms, Numer. Math. 105 (2006), no. 2, 193–216. MR 2262756, DOI 10.1007/s00211-006-0041-2
- Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
- Lucio Boccardo, Andrea Dall’Aglio, Thierry Gallouët, and Luigi Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1997), no. 1, 237–258. MR 1453181, DOI 10.1006/jfan.1996.3040
- L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), pp. 149-169.
- Eduardo Casas, $L^2$ estimates for the finite element method for the Dirichlet problem with singular data, Numer. Math. 47 (1985), no. 4, 627–632. MR 812624, DOI 10.1007/BF01389461
- Eduardo Casas, Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim. 35 (1997), no. 4, 1297–1327. MR 1453300, DOI 10.1137/S0363012995283637
- K. Chrysafinos and L. S. Hou, Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions, SIAM J. Numer. Anal. 40 (2002), no. 1, 282–306. MR 1921920, DOI 10.1137/S0036142900377991
- P. G. Ciarlet, The finite element methods for elliptic problems, North-Holland, Amsterdam, 1978.
- Klaus Deckelnick and Michael Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal. 45 (2007), no. 5, 1937–1953. MR 2346365, DOI 10.1137/060652361
- Klaus Deckelnick and Michael Hinze, Variational discretization of parabolic control problems in the presence of pointwise state constraints, J. Comput. Math. 29 (2011), no. 1, 1–15. MR 2723983, DOI 10.4208/jcm.1006-m3213
- J. Droniou and J.-P. Raymond, Optimal pointwise control of semilinear parabolic equations, Nonlinear Anal. 39 (2000), no. 2, Ser. A: Theory Methods, 135–156. MR 1722110, DOI 10.1016/S0362-546X(98)00170-9
- Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in $L_\infty L_2$ and $L_\infty L_\infty$, SIAM J. Numer. Anal. 32 (1995), no. 3, 706–740. MR 1335652, DOI 10.1137/0732033
- Donald A. French and J. Thomas King, Analysis of a robust finite element approximation for a parabolic equation with rough boundary data, Math. Comp. 60 (1993), no. 201, 79–104. MR 1153163, DOI 10.1090/S0025-5718-1993-1153163-1
- R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, Acta numerica, 1995, Acta Numer., Cambridge Univ. Press, Cambridge, 1995, pp. 159–333. MR 1352473, DOI 10.1017/s0962492900002543
- W. Gong and M. Hinze, Error estimates for parabolic optimal control problems with control and state constraints, preprint, Hamburger Beiträge zur Angewandten Mathematik, 2010-13.
- Wei Gong and Ningning Yan, A mixed finite element scheme for optimal control problems with pointwise state constraints, J. Sci. Comput. 46 (2011), no. 2, 182–203. MR 2753242, DOI 10.1007/s10915-010-9392-z
- R. Li and W. B. Liu, AFEPack available online at http://circus.math.pku.edu.cn/AFEPack.
- J.-L. Lions, Optimal control of systems governed by partial differential equations. , Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971. Translated from the French by S. K. Mitter. MR 0271512
- J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.
- Mitchell Luskin and Rolf Rannacher, On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal. 19 (1982), no. 1, 93–113. MR 646596, DOI 10.1137/0719003
- A. Martínez, C. Rodríguez, and M. E. Vázquez-Méndez, Theoretical and numerical analysis of an optimal control problem related to wastewater treatment, SIAM J. Control Optim. 38 (2000), no. 5, 1534–1553. MR 1766429, DOI 10.1137/S0363012998345640
- Dominik Meidner, Rolf Rannacher, and Boris Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time, SIAM J. Control Optim. 49 (2011), no. 5, 1961–1997. MR 2837507, DOI 10.1137/100793888
- Ricardo H. Nochetto and Claudio Verdi, Convergence past singularities for a fully discrete approximation of curvature-driven interfaces, SIAM J. Numer. Anal. 34 (1997), no. 2, 490–512. MR 1442924, DOI 10.1137/S0036142994269526
- Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. MR 1299729
- A. M. Ramos, R. Glowinski, and J. Periaux, Pointwise control of the Burgers equation and related Nash equilibrium problems: computational approach, J. Optim. Theory Appl. 112 (2002), no. 3, 499–516. MR 1892233, DOI 10.1023/A:1017907930931
- Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661, DOI 10.1090/S0025-5718-1982-0645661-4
- Alfred H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates, Math. Comp. 67 (1998), no. 223, 877–899. MR 1464148, DOI 10.1090/S0025-5718-98-00959-4
- Ridgway Scott, Finite element convergence for singular data, Numer. Math. 21 (1973/74), 317–327. MR 337032, DOI 10.1007/BF01436386
- Ridgway Scott, Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681–697. MR 436617, DOI 10.1090/S0025-5718-1976-0436617-2
- Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR 2249024
Additional Information
- Wei Gong
- Affiliation: Schwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany
- Address at time of publication: LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
- Email: wgong@lsec.cc.ac.cn
- Received by editor(s): February 5, 2011
- Received by editor(s) in revised form: August 8, 2011, and September 13, 2011
- Published electronically: August 8, 2012
- Additional Notes: This work was partially supported by the National Natural Science Foundation of China under grant 11171337 and the National Basic Research Program of China under grant 2012 CB821204
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 82 (2013), 69-98
- MSC (2010): Primary 49J20, 49K20, 65N15, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2012-02630-5
- MathSciNet review: 2983016