Previous |  Up |  Next


elliptic equation; nonlinear Newton boundary condition; weak solution; finite element discretization; numerical integration; error estimation; effect of numerical integration
This paper is concerned with the analysis of the finite element method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained by numerical integration. The theoretical results are verified by numerical experiments.\looseness -1
[1] M. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, G. N. Wells: The FEniCS Project Version 1.5. Archive of Numerical Software 3 (2015), 9-23. DOI 10.11588/ans.2015.100.20553
[2] Bialecki, R., Nowak, A. J.: Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions. Appl. Math. Modelling 5 (1981), 417-421. DOI 10.1016/S0307-904X(81)80024-8 | Zbl 0475.65078
[3] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North Holland, Amsterdam (1978). DOI 10.1016/S0168-2024(08)70178-4 | MR 0520174 | Zbl 0383.65058
[4] Dolejší, V., Feistauer, M.: Discontinuous Galerkin Method. Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics 48, Springer, Cham (2015). DOI 10.1007/978-3-319-19267-3 | MR 3363720 | Zbl 1401.76003
[5] J. Douglas, Jr., T. Dupont: Galerkin methods for parabolic equations with nonlinear boundary conditions. Numer. Math. 20 (1973), 213-237. DOI 10.1007/BF01436565 | MR 0319379 | Zbl 0234.65096
[6] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics 19, American Mathematical Society, Providence (2010). DOI 10.1090/gsm/019 | MR 2597943 | Zbl 1194.35001
[7] Feistauer, M., Kalis, H., Rokyta, M.: Mathematical modelling of an electrolysis process. Commentat. Math. Univ. Carol. 30 (1989), 465-477. MR 1031864 | Zbl 0704.35021
[8] Feistauer, M., Najzar, K.: Finite element approximation of a problem with a nonlinear Newton boundary condition. Numer. Math. 78 (1998), 403-425. DOI 10.1007/s002110050318 | MR 1603350 | Zbl 0888.65118
[9] Feistauer, M., Najzar, K., Sobotíková, V.: Error estimates for the finite element solution of elliptic problems with nonlinear Newton boundary conditions. Numer. Funct. Anal. Optimization 20 (1999), 835-851. DOI 10.1080/01630569908816927 | MR 1728186 | Zbl 0947.65116
[10] Feistauer, M., Roskovec, F., Sändig, A.-M.: Discontinuous Galerkin method for an elliptic problem with nonlinear Newton boundary conditions in a polygon. IMA J. Numer. Anal. 39 (2019), (423-453). DOI 10.1093/imanum/drx070 | MR 3903559
[11] Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge Monographs on Applied and Computational Mathematics 1, Cambridge University Press, Cambridge (1996). DOI 10.1017/CBO9780511626357 | MR 1386891 | Zbl 0844.65084
[12] Franců, J.: Monotone operators. A survey directed to applications to differential equations. Appl. Mat. 35 (1990), 257-301. MR 1065003 | Zbl 0724.47025
[13] Ganesh, M., Graham, I. G., Sivaloganathan, J.: A pseudospectral three-dimensional boundary integral method applied to a nonlinear model problem from finite elasticity. SIAM J. Numer. Anal. 31 (1994), 1378-1414. DOI 10.1137/0731072 | MR 1293521 | Zbl 0815.41008
[14] Ganesh, M., Steinbach, O.: Nonlinear boundary integral equations for harmonic problems. J. Integral Equations Appl. 11 (1999), 437-459. DOI 10.1216/jiea/1181074294 | MR 1738277 | Zbl 0974.65112
[15] Ganesh, M., Steinbach, O.: Boundary element methods for potential problems with nonlinear boundary conditions. Math. Comput. 70 (2001), 1031-1042. DOI 10.1090/S0025-5718-00-01266-7 | MR 1826575 | Zbl 0971.65107
[16] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics 24, Pitman Publishing, Boston (1985). DOI 10.1137/1.9781611972030 | MR 0775683 | Zbl 0695.35060
[17] Harriman, K., Houston, P., Senior, B., Süli, E.: $hp$-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. Recent Advances in Scientific Computing and Partial Differential Equations 2002 S. Y. Cheng et al. Contemp. Math. 330; American Mathematical Society, Providence (2003), 89-119. DOI 10.1090/conm/330 | MR 2011714 | Zbl 1037.65117
[18] Hesthaven, J. S.: From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35 (1998), 655-676. DOI 10.1137/S003614299630587X | MR 1618874 | Zbl 0933.41004
[19] Křížek, M., Liu, L., Neittaanmäki, P.: Finite element analysis of a nonlinear elliptic problem with a pure radiation condition. Applied Nonlinear Analysis A. Sequeira et al. Kluwer Academic/Plenum Publishers, New York (1999), 271-280. DOI 10.1007/0-306-47096-9_19 | MR 1727454 | Zbl 0953.65081
[20] Liu, L., Křížek, M.: Finite element analysis of a radiation heat transfer problem. J. Comput. Math. 16 (1998), 327-336. MR 1640963 | Zbl 0919.65067
[21] Moreau, R., Ewans, J. W.: An analysis of the hydrodynamics of aluminum reduction cells. J. Electrochem. Soc. 31 (1984), 2251-2259. DOI 10.1149/1.2115235
[22] Powell, M. J. D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981). DOI 10.1017/cbo9781139171502 | MR 0604014 | Zbl 0453.41001
[23] Rack, H.-J., Vajda, R.: Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant. Stud. Univ. Babeş-Bolyai, Math. 60 (2015), 151-171. MR 3355978 | Zbl 1374.05134
[24] Roubíček, T.: A finite-element approximation of Stefan problems in heterogeneous media. Free Boundary Value Problems 1989 Int. Ser. Numer. Math. 95, Birkhäuser, Basel (1990), 267-275. DOI 10.1007/978-3-0348-7301-7_16 | MR 1111033 | Zbl 0721.65081
[25] Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987). MR 0924157 | Zbl 0925.00005
Partner of
EuDML logo