Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data

Eymard, Robert; Maltese, David

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 6, pp. 3043-3089

This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods show convergence properties to a continuous solution of the problem in a weak sense, through the change of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and ℝ, and v is valued in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D and 3D cases where the solution does not belong to H¹ (Ω), show that this method can provide accurate results. We then construct a numerical scheme which presents a convergence property to the entropy weak solution of the problem in the case where the right-hand side belongs to L¹. This is achieved owing to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation, and adding an upstream weighting evaluation and a nonlinear p-Laplace vanishing stabilisation term.

Classification : 65N12, 65N30
Keywords: Elliptic equations with irregular data, finite elements, control-volume finite elements, entropy solution

[1] R. A. Adams, Sobolev Spaces. Pure and Applied Mathematics. Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). | MR | Zbl

[2] A. Ait Hammou Oulhaj, C. Cancès and C. Chainais-Hillairet, Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy. ESAIM: M2AN 52 (2018) 1533–1567. | MR | Zbl | Numdam | DOI

[3] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995) 241–273. | MR | Zbl | Numdam

[4] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989) 149–169. | MR | Zbl | DOI

[5] L. Boccardo, T. Gallouët and J. L. Vázquez, Nonlinear elliptic equations in ${𝐑}^N$ without growth restrictions on the data. J. Differ. Equ. 105 (1993) 334–363. | MR | Zbl | DOI

[6] C. Cancès and C. Guichard, Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations. Math. Comp. 85 (2016) 549–580. | MR | Zbl | DOI

[7] C. Cancès, M. Ibrahim and M. Saad, Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system. SMAI J. Comput. Math. 3 (2017) 1–28. | MR | Zbl | DOI

[8] J. Casado-Daz, T. Chacón Rebollo, V. Girault, M. Gómez Mármol and F. Murat, Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in $L^1$. Numer. Math. 105 (2007) 337–374. | MR | Zbl | DOI

[9] P. G. Ciarlet, The finite element method for elliptic problems. In: Studies in Mathematics and its Applications. Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford (1978) xix+530. | MR | Zbl

[10] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999) 741–808. | MR | Zbl | Numdam

[11] A. Dall’Aglio, A remark on the entropy solutions. Personal Communication (1996).

[12] K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, Berlin (1985). | MR | Zbl | DOI

[13] J. Droniou, Solving convection-diffusion equations with mixed, neumann and fourier boundary conditions and measures as data, by a duality method. Adv. Differ. Equ. 5 (2000) 1341–1396. | MR | Zbl

[14] J. Droniou, Non-coercive linear elliptic problems. Potential Anal. 17 (2002) 181–203. | MR | Zbl | DOI

[15] J. Droniou, T. Gallouët and R. Herbin, A finite volume scheme for a noncoercive elliptic equation with measure data. SIAM J. Numer. Anal. 41 (2003) 1997–2031. | MR | Zbl | DOI

[16] A. Ern and J.-L. Guermond, Applied Mathematical Sciences. Springer-Verlag, New York (2004). | DOI

[17] R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009–1043. | MR | Zbl | DOI

[18] R. Eymard, T. Gallouët, C. Guichard, R. Herbin and R. Masson, TP or not TP, that is the question. Comput. Geosci. 18 (2014) 285–296. | MR | Zbl | DOI

[19] P. Fabrie and T. Gallouët, Modelling wells in porous media flow. Math. Models Methods Appl. Sci. 10 (2000) 673–709. | MR | Zbl | DOI

[20] T. Gallouët and R. Herbin, Existence of a solution to a coupled elliptic system. Appl. Math. Lett. 7 (1994) 49–55. | MR | Zbl | DOI

[21] T. Gallouët and A. Monier, On the regularity of solutions to elliptic equations. Rend. Mat. Appl. 19 (1999) 471–488. | MR | Zbl

[22] N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963) 189–206. | MR | Zbl | Numdam

[23] A. Prignet, Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures. Rend. Mat. Appl. 15 (1995) 321–337. | MR | Zbl

[24] J. Serrin, Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964) 385–387. | MR | Zbl | Numdam

[25] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. | MR | Zbl | Numdam | DOI