Open Access
Issue
ESAIM: M2AN
Volume 57, Number 2, March-April 2023
Page(s) 585 - 620
DOI https://doi.org/10.1051/m2an/2022095
Published online 27 March 2023
  1. R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Elsevier (2003). [Google Scholar]
  2. S. Ariche, C. De Coster and S. Nicaise, Regularity of solutions of elliptic problems with a curved fracture. J. Math. Anal. App. 447 (2017) 908–932. [Google Scholar]
  3. S. Bertoluzza, A. Decoene, L. Lacouture and S. Martin, Local error estimates of the finite element method for an elliptic problem with a Dirac source term. Numer. Methods Part. Differ. Equ. 34 (2018) 97–120. [Google Scholar]
  4. S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods. Vol. 15. Springer Science & Business Media (2007). [Google Scholar]
  5. E. Casas, L2 estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math. 47 (1985) 627–632. [Google Scholar]
  6. L. Cattaneo and P. Zunino, A computational model of drug delivery through microcirculation to compare different tumor treatments. Int. J. Numer. Methods Biomed. Eng. 30 (2014) 1347–1371. [CrossRef] [MathSciNet] [Google Scholar]
  7. Z. Chen and H. Chen, Pointwise error estimates of discontinuous Galerkin methods with penalty for second-order elliptic problems. SIAM J. Numer. Anal. 42 (2004) 1146–1166. [Google Scholar]
  8. W. Choi and S. Lee, Optimal error estimate of elliptic problems with Dirac sources for discontinuous and enriched Galerkin methods. Appl. Numer. Math. 150 (2020) 76–104. [CrossRef] [MathSciNet] [Google Scholar]
  9. K. Chrysafinos and L. Steven Hou, Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions. SIAM J. Numer. Anal. 40 (2002) 282–306. [Google Scholar]
  10. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. SIAM (2002). [Google Scholar]
  11. C. D’Angelo, Multiscale modelling of metabolism and transport phenomena in living tissues. Technical report, EPFL (2007). [Google Scholar]
  12. 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. [Google Scholar]
  13. C. D’Angelo and 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. [Google Scholar]
  14. M.C. Delfour and J.-P. Zolésio, Shape analysis via oriented distance functions. J. Funct. Anal. 123 (1994) 129–201. [Google Scholar]
  15. D. Di Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Math. Comput. 79 (2010) 1303–1330. [Google Scholar]
  16. I. Drelichman, R.G. Durán and I. Ojea, A weighted setting for the numerical approximation of the Poisson problem with singular sources. SIAM J. Numer. Anal. 58 (2020) 590–606. [Google Scholar]
  17. R.G. Durán and F. López Garca, Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains. Math. Models Methods Appl. Sci. 20 (2010) 95–120. [Google Scholar]
  18. L.C. Evans, Partial Differential Equations. American Mathematical Society (2010). [Google Scholar]
  19. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer (2015). [Google Scholar]
  20. I.G. Gjerde, K. Kumar, J.M. Nordbotten and B. Wohlmuth, Splitting method for elliptic equations with line sources. ESAIM: Math. Modell. Numer. Anal. 53 (2019) 1715–1739. [CrossRef] [EDP Sciences] [Google Scholar]
  21. I.G. Gjerde, K. Kumar and J.M. Nordbotten, A singularity removal method for coupled 1D–3D flow models. Comput. Geosci. 24 (2020) 443–457. [CrossRef] [MathSciNet] [Google Scholar]
  22. W. Gong, Error estimates for finite element approximations of parabolic equations with measure data. Math. Comput. 82 (2013) 69–98. [Google Scholar]
  23. W. Gong and N. Yan, Finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold. SIAM J. Numer. Anal. 54 (2016) 1229–1262. [Google Scholar]
  24. W. Gong, G. Wang and N. Yan, Approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold. SIAM J. Control Optim. 52 (2014) 2008–2035. [Google Scholar]
  25. P. Houston and T.P. Wihler, Discontinuous Galerkin methods for problems with Dirac delta source. ESAIM: Math. Modell. Numer. Anal. – Modél. Math. Anal. Numér. 46 (2012) 1467–1483. [CrossRef] [EDP Sciences] [Google Scholar]
  26. T. Köppl and B. Wohlmuth, Optimal a priori error estimates for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 52 (2014) 1753–1769. [Google Scholar]
  27. T. Köppl, E. Vidotto and B. Wohlmuth, A local error estimate for the Poisson equation with a line source term, in Numerical Mathematics and Advanced Applications ENUMATH 2015. Springer (2016) 421–429. [Google Scholar]
  28. H. Leng and Y. Chen, A hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac delta source. ESAIM: Math. Modell. Numer. Anal. 56 (2022) 385–406. [CrossRef] [EDP Sciences] [Google Scholar]
  29. P.A. Nguyen and J.-P. Raymond, Control problems for convection-diffusion equations with control localized on manifolds. ESAIM: Control Optim. Calculus Variations 6 (2001) 467–488. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  30. J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods. Math. Comput. 28 (1974) 937–958. [Google Scholar]
  31. R.H. Nochetto, E. Otárola and A.J. Salgado, Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numer. Math. 132 (2016) 85–130. [Google Scholar]
  32. I. Ojea, Optimal a priori error estimates in weighted Sobolev spaces for the Poisson problem with singular sources. ESAIM: Math. Modell. Numer. Anal. 55 (2021) S879–S907. [CrossRef] [EDP Sciences] [Google Scholar]
  33. B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM (2008). [CrossRef] [Google Scholar]
  34. R. Scott, Finite element convergence for singular data. Numer. Math. 21 (1973) 317–327. [Google Scholar]
  35. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Vol. 25. Springer Science & Business Media (2007). [Google Scholar]
  36. L.B. Wahlbin, Local behavior in finite element methods. Handb. Numer. Anal. 2 (1991) 353–522. [Google Scholar]
  37. C. Waluga and B. Wohlmuth, Quasi-optimal a priori interface error bounds and a posteriori estimates for the interior penalty method. SIAM J. Numer. Anal. 51 (2013) 3259–3279. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you