We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is quasi-optimal in two-dimensional space and suboptimal in three-dimensional space. Numerical simulations are provided to confirm our findings.


  1. finite elements
  2. interface problems
  3. immersed boundary method
  4. Dirac delta approximations
  5. a posteriori error estimates
  6. adaptivity

MSC codes

  1. 65N15
  2. 65N30
  3. 65N50

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The author would like to thank the anonymous reviewers who provided valuable comments and remarks on the earlier version of the manuscript.


A. Allendes, E. Otárola, and A. J. Salgado, A posteriori error estimates for the Stokes problem with singular sources, Comput. Methods Appl. Mech. Engrg., 345 (2019), pp. 1007–1032.
G. Alzetta and L. Heltai, Multiscale modeling of fiber reinforced materials via non-matching immersed methods, Comput. Structures, 239 (2020), 106334.
D. Arndt, W. Bangerth, B. Blais, M. Fehling, R. Gassmöller, T. Heister, L. Heltai, U. Köcher, M. Kronbichler, M. Maier, et al., The deal.II library, version 9.3, J. Numer. Math., 29 (2021), pp. 171–186.
D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin, and D. Wells, The deal.II finite element library: Design, features, and insights, Comput. Math. Appl., 81 (2021), pp. 407–422.
R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283–301.
S. Berrone, A. Bonito, R. Stevenson, and M. Verani, An optimal adaptive fictitious domain method, Math. Comp., 88 (2019), pp. 2101–2134.
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 Partial Differential Equations, 34 (2017), pp. 97–120.
P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), pp. 219–268.
D. Boffi, F. Credali, and L. Gastaldi, On the interface matrix for fluid-structure interaction problems with fictitious domain approach, Comput. Methods Appl. Mech. Engrg., 401, (2022), 115650.
D. Boffi and L. Gastaldi, A finite element approach for the immersed boundary method, Comput. Structures, 81 (2003), pp. 491–501.
D. Boffi and L. Gastaldi, On the existence and the uniqueness of the solution to a fluid-structure interaction problem, J. Differential Equations, 279 (2021), pp. 136–161.
A. Bonito, R. A. DeVore, and R. H. Nochetto, Adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Numer. Anal., 51 (2013), pp. 3106–3134.
A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), pp. 734–771.
J. M. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524–2550.
D. Cerroni, F. Laurino, and P. Zunino, Mathematical analysis, finite element approximation and numerical solvers for the interaction of 3D reservoirs with 1D wells, GEM Int. J. Geomath., 10 (2019).
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Appl. Math. 40, SIAM, Philadelphia, PA, 2002.
A. Cohen, R. DeVore, and R. H. Nochetto, Convergence rates of AFEM with \(H^{-1}\) data, Found. Comput. Math., 12 (2012), pp. 671–718.
W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124.
W. Dörfler and R. H. Nochetto, Small data oscillation implies the saturation assumption, Numer. Math., 91 (2002), pp. 1–12.
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer-Verlag, New York, 2004.
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics 24, Pitman, Boston, MA, 1985.
L. Heltai and A. Caiazzo, Multiscale modeling of vascularized tissues via nonmatching immersed methods, Int. J. Numer. Methods Biomed. Eng., 35 (2019), e3264.
L. Heltai, A. Caiazzo, and L. O. Müller, Multiscale coupling of one-dimensional vascular models and elastic tissues, Ann. Biomed. Eng., 49 (2021), pp. 3243–3254.
L. Heltai and F. Costanzo, Variational implementation of immersed finite element methods, Comput. Methods Appl. Mech. Engrg., 229 (2012), pp. 110–127.
L. Heltai and W. Lei, A priori error estimates of regularized elliptic problems, Numer. Math., 146 (2020), pp. 571–596.
L. Heltai and W. Lei, Adaptive Finite Element Approximations for Elliptic Problems Using Regularized Forcing Data, preprint, https://arxiv.org/abs/2110.15029, 2021.
L. Heltai and N. Rotundo, Error estimates in weighted sobolev norms for finite element immersed interface methods, Comput. Math. Appl., 78 (2019), pp. 3586–3604.
B. Hosseini, N. Nigam, and J. M. Stockie, On regularizations of the Dirac delta distribution, J. Comput. Phys., 305 (2016), pp. 423–447.
P. Houston and T. P. Wihler, Discontinuous galerkin methods for problems with Dirac delta source, ESAIM Math. Model. Numer. Anal., 46 (2012), pp. 1467–1483.
R. Krause and P. Zulian, A parallel approach to the variational transfer of discrete fields between arbitrarily distributed unstructured finite element meshes, SIAM J. Sci. Comput., 38 (2016), pp. C307–C333.
C. Kreuzer and A. Veeser, Oscillation in a posteriori error estimation, Numer. Math., 148 (2021), pp. 43–78.
H. Li, X. Wan, P. Yin, and L. Zhao, Regularity and finite element approximation for two-dimensional elliptic equations with line dirac sources, J. Comput. Appl. Math., 393 (2021), 113518.
F. Millar, I. Muga, S. Rojas, and K. G. V. der Zee, Projection in Negative Norms and the Regularization of Rough Linear Functionals, https://arxiv.org/abs/arXiv:2101.03044, 2021.
W. F. Mitchell, A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software, 15 (1989), pp. 326–347, 1990.
R. Mittal and G. Iaccarino, Immersed boundary methods, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech. 37, Annual Reviews, Palo Alto, CA, 2005, pp. 239–261.
P. Morin, R. H. Nochetto, and K. G. Siebert, Data oscillation and convergence of adaptive fem, SIAM J. Numer. Anal., 38 (2000), pp. 466–488.
R. H. Nochetto, Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comp., 64 (1995), pp. 1–22.
R. H. Nochetto, K. G. Siebert, and A. Veeser, Theory of adaptive finite element methods: An introduction, in Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin, 2009, pp. 409–542.
C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), pp. 479–517.
D. D. Silva, F. Ferrari, and S. Salsa, Perron’s solutions for two-phase free boundary problems with distributed sources, Nonlinear Anal. Theory Methods Appl., 121 (2015), pp. 382–402.
R. Stevenson, An optimal adaptive finite element method, SIAM J. Numer. Anal., 42 (2005), pp. 2188–2217.
R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), pp. 245–269.
R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77 (2008), pp. 227–241.
J.-P. Suarez, G. B. Jacobs, and W.-S. Don, A high-order Dirac-delta regularization with optimal scaling in the spectral solution of one-dimensional singular hyperbolic conservation laws, SIAM J. Sci. Comput., 36 (2014), pp. A1831–A1849.
A.-K. Tornberg, Multi-dimensional quadrature of singular and discontinuous functions, BIT, 42 (2002), pp. 644–669.

Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 431 - 456
ISSN (online): 1095-7170


Submitted: 28 October 2021
Accepted: 25 October 2022
Published online: 10 March 2023


  1. finite elements
  2. interface problems
  3. immersed boundary method
  4. Dirac delta approximations
  5. a posteriori error estimates
  6. adaptivity

MSC codes

  1. 65N15
  2. 65N30
  3. 65N50



Luca Heltai
Mathematics Area, SISSA – International School for Advanced Studies, via Bonomea 265, 34136, Trieste, Italy.
School of Mathematical Sciences, University of Electronic Science and Technology of China, No2006, Xiyuan Ave, West Hi-Tech Zone, 611731, Chengdu, China.

Funding Information

Italian Ministry of Education, University, and Research
Funding: The work of the authors was partially supported by the National Research Projects (PRIN 2017) “Numerical Analysis for Full and Reduced Order Methods for the Efficient and Accurate Solution of Complex Systems Governed by Partial Differential Equations,” funded by the Italian Ministry of Education, University, and Research.

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