A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces

Juan Pablo Agnelli; Eduardo M. Garau; Pedro Morin

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 6, page 1557-1581
  • ISSN: 0764-583X

Abstract

top
In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation operators, without need of modifications, and makes use of weighted estimates for fractional integrals and maximal functions. Numerical experiments with an adaptive algorithm yield optimal meshes and very good effectivity indices.

How to cite

top

Agnelli, Juan Pablo, Garau, Eduardo M., and Morin, Pedro. "A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.6 (2014): 1557-1581. <http://eudml.org/doc/273164>.

@article{Agnelli2014,
abstract = {In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation operators, without need of modifications, and makes use of weighted estimates for fractional integrals and maximal functions. Numerical experiments with an adaptive algorithm yield optimal meshes and very good effectivity indices.},
author = {Agnelli, Juan Pablo, Garau, Eduardo M., Morin, Pedro},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {elliptic problems; point sources; a posteriori error estimates; finite elements; weighted Sobolev spaces; a posteriori error estimates},
language = {eng},
number = {6},
pages = {1557-1581},
publisher = {EDP-Sciences},
title = {A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces},
url = {http://eudml.org/doc/273164},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Agnelli, Juan Pablo
AU - Garau, Eduardo M.
AU - Morin, Pedro
TI - A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 6
SP - 1557
EP - 1581
AB - In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation operators, without need of modifications, and makes use of weighted estimates for fractional integrals and maximal functions. Numerical experiments with an adaptive algorithm yield optimal meshes and very good effectivity indices.
LA - eng
KW - elliptic problems; point sources; a posteriori error estimates; finite elements; weighted Sobolev spaces; a posteriori error estimates
UR - http://eudml.org/doc/273164
ER -

References

top
  1. [1] Th. Apel, O. Benedix, D. Sirch and B. Vexler, A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal.49 (2011) 992–1005. Zbl1229.65203MR2812554
  2. [2] Th. Apel, A.-M. Sändig, J.R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci.19 (1996) 63–85. Zbl0838.65109MR1365264
  3. [3] R. Araya, E. Behrens, R. Rodríguez, An adaptive stabilized finite element scheme for a water quality model. Comput. Methods Appl. Mech. Engrg.196 (2007) 2800–2812. Zbl1123.76027MR2325392
  4. [4] R. Araya, E. Behrens, R. Rodríguez, A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math.105 (2006) 193–216. Zbl1162.65401MR2262756
  5. [5] I. Babuška, Error-Bounds for Finite Element Method. Numer. Math.16 (1971) 322–333. Zbl0214.42001MR288971
  6. [6] I. Babuška, M.B. Rosenzweig, A finite element scheme for domains with corners. Numer. Math. 20 (1972/73) 1–21. Zbl0252.65084MR323129
  7. [7] Z. Belhachmi, C. Bernardi, S. Deparis, Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. Numer. Math.105 (2006) 217–247. Zbl1107.65103MR2262757
  8. [8] E. Casas, L2-estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math.47 (1985) 627–632. Zbl0561.65071MR812624
  9. [9] P. Clément, Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique9 (1975) 77–84. Zbl0368.65008MR400739
  10. [10] C. D’Angelo, Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: Applications to one- and three-dimensional coupled problems. SIAM J. Numer. Anal.50 (2012) 194–215. Zbl1246.65215MR2888310
  11. [11] C. D’Angelo, A. Quarteroni, On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18 (2008) 1481–1504. Zbl05360522MR2439847
  12. [12] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal.33 (1996) 1106–1124. Zbl0854.65090MR1393904
  13. [13] K. Eriksson, Improved accuracy by adapted mesh-refinements in the finite element method. Math. Comput.44 (1985) 321–343. Zbl0571.65097MR777267
  14. [14] L.C. Evans, Partial diferential equations. Grad. Stud. Math., vol. 19. American Mathematical Society, Providence, RI (1998). Zbl0902.35002MR1625845
  15. [15] E. Fabes, C. Kenig, R. Serapioni, The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equations7 (1982) 77–116. Zbl0498.35042MR643158
  16. [16] F. Gaspoz, P. Morin, A. Veeser, A posteriori error estimates with point sources, in preparation (2013). 
  17. [17] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224. Springer-Verlag, Berlin (1983). Zbl0562.35001MR737190
  18. [18] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations. Oxford Science Publications (1993). Zbl1115.31001MR1207810
  19. [19] T. Kilpeläinen, Smooth approximation in weighted Sobolev spaces, Comment. Math. Univ. Carolinae38 (1997) 29–35. Zbl0886.46035
  20. [20] V.A. Kozlov, V.G. Maz’ya, J. Rossmann, Elliptic boundary value problems in domains with point singularities, Math. Surv. Monogr., vol. 52. American Mathematical Society, Providence, RI (1997). Zbl0947.35004MR1469972
  21. [21] A. Kufner, Weighted Sobolev spaces, A Wiley-Interscience Publication. John Wiley & Sons Inc., New York (1985). Translated from the Czech. Zbl0567.46009MR802206
  22. [22] P. Morin, K.G. Siebert, A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci.18 (2008) 707–737. Zbl1153.65111MR2413035
  23. [23] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc.165 (1972) 207–226. Zbl0236.26016MR293384
  24. [24] B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc.192 (1974) 261–274. Zbl0289.26010MR340523
  25. [25] J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. III. Ser. 16 (1962) 305–326. Zbl0112.33101MR163054
  26. [26] R.H. Nochetto, K.G. Siebert, A. Veeser, Theory of adaptive finite element methods: an introduction. Edited by R.A. DeVore, A. Kunoth. Multiscale, nonlinear and adaptive approximation. Springer, Berlin (2009) 409–542. Zbl1190.65176MR2648380
  27. [27] B. Opic, A. Kufner, Hardy-type inequalities, Pitman Res. Notes Math. Ser., vol. 219. Longman Scientific & Technical, Harlow (1990). Zbl0698.26007MR1069756
  28. [28] L.R. Scott, Finite Element Convergence for Singular Data. Numer. Math.21 (1973) 317–327. Zbl0255.65037MR337032
  29. [29] K.G. Siebert, A convergence proof for adaptive finite elements without lower bound. IMA J. Numer. Anal.31 (2011) 947–970. Zbl1225.65101MR2832786
  30. [30] L.R. Scott, S. Zhang, Finite Element Interpolation of Nonsmooth Functions Satisfying Boundary Conditions. Math. Comput.54 (1990) 483–493. Zbl0696.65007MR1011446
  31. [31] T.I. Seidman, M.K. Gobbert, D.W. Trott, M. Kružík, Finite element approximation for time-dependent diffusion with measure-valued source. Numer. Math.122 (2012) 709–723. Zbl1266.65172MR2995178

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.