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Keywords:
Elliptic equation, nonlinear Newton boundary condition, monotone operator method, finite element method, discontinuous Galerkin method, regularity and singular behaviour of the solution, error estimation
Elliptic equation, nonlinear Newton boundary condition, monotone operator method, finite element method, discontinuous Galerkin method, regularity and singular behaviour of the solution, error estimation
Summary:
The paper is concerned with the numerical analysis of an elliptic equation in a polygon with a nonlinear Newton boundary condition, discretized by the finite element or discontinuous Galerkin methods. Using the monotone operator theory, it is possible to prove the existence and uniqueness of the exact weak solution and the approximate solution. The main attention is paid to the study of error estimates. To this end, the regularity of the weak solution is investigated and it is shown that due to the boundary corner points, the solution looses regularity in a vicinity of these points. It comes out that the error estimation depends essentially on the opening angle of the corner points and on the parameter defining the nonlinear behaviour of the Newton boundary condition. Theoretical results are compared with numerical experiments confirming a nonstandard behaviour of error estimates.
References:
[1] Aln{\ae}s, M.M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS Project Version 1.5. Archive of Numerical Software, 2015.
[2] Babuška, I.: Private communication. Austin 2017.
[3] Bartoš, O.: Discontinuous Galerkin method for the solution of boundary-value problems in nonsmooth domains. Master Thesis, Faculty of Mathematics and Physics, Charles University, Praha 2017.
[4] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. MR 0520174 | Zbl 0547.65072
[5] Feistauer, M., Roskovec, F., Sandig, A.-M.: Discontinuous Galerkin method for an elliptic problem with nonlinear Newton boundary conditions in a polygon. IMA J. Numer. Anal. (to appear). MR 3903559
[6] Ganesh, M., Steinbach, O.: Boundary element methods for potential problems with nonlinear boundary conditions. Mathematics of Computation 70 (2000), 1031–1042. DOI 10.1090/S0025-5718-00-01266-7 | MR 1826575
[7] Ganesh, M., Steinbach, O.: Nonlinear boundary integral equations for harmonic problems. Journal of Integral Equations and Applications 11 (1999), 437–459. DOI 10.1216/jiea/1181074294 | MR 1738277
[8] Harriman, K., Houston, P., Senior, B., Suli, E.: hp-Version Discontinuous Galerkin Methods with Interior Penalty for Partial Differential Equations with Nonnegative Characteristic Form. Contemporary Mathematics Vol. 330, pp. 89-119, AMS, 2003. DOI 10.1090/conm/330/05886 | MR 2011714
[9] Křížek, M., Liu, L., Neittaanmäki, P.: Finite element analysis of a nonlinear elliptic problem with a pure radiation condition. In: Proc. Conf. devoted to the 70th birthday of Prof. J. Nečas, Lisbon, 1999. MR 1727454
[10] Liu, L., Křížek, M.: Finite element analysis of a radiation heat transfer problem. J. Comput. Math. 16 (1998), 327–336.
[11] Moreau, R., Ewans, J. W.: An analysis of the hydrodynamics of alluminium reduction cells. J. Electrochem. Soc. 31 (1984), 2251–2259. DOI 10.1149/1.2115235
[12] Pick, L., Kufner, A., John, O., Fučík, S.: Function Spaces. De Gruyter Series in Nonlinear Analysis and Applications 14, Berlin, 2013. MR 3024912
[13] Rudin, W.: Real and comples analysis. McGraw-Hill, 1987. MR 0924157
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