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Keywords:
elliptic equation; nonlinear Newton boundary condition; monotone operator method; finite element approximation; approximation of a curved boundary; numerical integration; ideal triangulation; ideal interpolation; convergence of the finite element method
Summary:
The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional nonpolygonal domain with a curved boundary. The existence and uniqueness of the solution of the continuous problem is a consequence of the monotone operator theory. The main attention is paid to the effect of the basic finite element variational crimes: approximation of the curved boundary by a polygonal one and the evaluation of integrals by numerical quadratures. With the aid of some important properties of Zlámal’s ideal triangulation and interpolation, the convergence of the method is analyzed.
References:
[1] R.  Bialecki, A. J.  Nowak: 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
[2] S. S. Chow: Finite element error estimates for nonlinear elliptic equations of monotone type. Numer. Math. 54 (1988), 373–393. MR 0972416 | Zbl 0643.65058
[3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[4] P. G.  Ciarlet, P. A. Raviart: The combined effect of curved boundaries and numerical integration in isoparametric finite element method. In: The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, A. K. Aziz (ed.), Academic Press, New York, 1972, pp. 409–474. MR 0421108
[5] M.  Feistauer: On the finite element approximation of a cascade flow problem. Numer. Math. 50 (1987), 655–684. DOI 10.1007/BF01398378 | MR 0884294 | Zbl 0646.76085
[6] M.  Feistauer, H. Kalis and M. Rokyta: Mathematical modelling of an electrolysis process. Comment. Math. Univ. Carolin. 30 (1989), 465–477. MR 1031864
[7] M. Feistauer, M.  Křížek and V.  Sobotíková: An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type. East-West J.  Numer. Math. 1 (1993), 267–285. MR 1318806
[8] M.  Feistauer, K.  Najzar: Finite element approximation of a problem with a nonlinear Newton boundary condition. Numer. Math. 78 (1998), 403–425. DOI 10.1007/s002110050318 | MR 1603350
[9] M.  Feistauer, K.  Najzar and V. Sobotíková: Error estimates for the finite element solution of elliptic problems with nonlinear Newton boundary conditions. Numer. Funct. Anal. Optim. 20 (1999), 835–851. DOI 10.1080/01630569908816927 | MR 1728186
[10] M.  Feistauer, K.  Najzar, V.  Sobotíková and P.  Sváček: Numerical analysis of problems with nonlinear Newton boundary conditions. In: Numerical Mathematics and Advanced Applications, Proc. of the Conf. ENUMATH99, P.  Neittaanmäki, T. Tiihonen and P. Tarvainen (eds.), World Scientific, Singapore, 2000, pp. 486–493.
[11] M.  Feistauer, V.  Sobotíková: Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. RAIRO Modél. Math. Anal. Numér. 24 (1990), 457–500. MR 1070966
[12] M.  Feistauer, A.  Ženíšek: Finite element solution of nonlinear elliptic problems. Numer. Math. 50 (1987), 451–475. DOI 10.1007/BF01396664 | MR 0875168
[13] M.  Feistauer, A.  Ženíšek: Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math. 52 (1988), 147–163. DOI 10.1007/BF01398687 | MR 0923708
[14] J. Franců: Monotone operators. A survey directed to applications to differential equations. Appl. Math. 35 (1990), 257–301. MR 1065003
[15] M.  Ganesh, I. G.  Graham and J.  Sivaloganathan: 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
[16] M. Ganesh, O. Steinbach: Boundary element methods for potential problems with nonlinear boundary conditions. Applied Mathematics Report AMR98/17, School of Mathematics, The University of New South Wales, Sydney (1998).
[17] M.  Ganesh, O.  Steinbach: Nonlinear boundary integral equations for harmonic problems. Applied Mathematics Report AMR98/20, School of Mathematics, The University of New South Wales, Sydney (1998). MR 1738277
[18] J.  Gwinner: A discretization theory for monotone semicoercive problems and finite element convergence for $p$-harmonic Signorini problems. Z. Angew. Math. Mech. 74 (1994), 417–427. DOI 10.1002/zamm.19940740917 | MR 1296460 | Zbl 0823.31006
[19] M.  Křížek, L.  Liu and P.  Neittaanmäki: Finite element analysis of a nonlinear elliptic problem with a pure radiation condition. In: Applied Nonlinear Analysis, Kluwer, Amsterdam, 1999, pp. 271–280. MR 1727454
[20] A.  Kufner, O.  John and S.  Fučík: Function Spaces. Academia, Praha, 1977. MR 0482102
[21] L.  Liu, M.  Křížek: Finite element analysis of a radiation heat transfer problem. J.  Comput. Math. 16 (1998), 327–336.
[22] R. Moreau, J. W.  Ewans: An analysis of the hydrodynamics of aluminium reduction cells. J.  Electrochem. Soc. 31 (1984), 2251–2259.
[23] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. MR 0227584
[24] V. Sobotíková: Finite elements on curved domains. East-West J.  Numer. Math. 4 (1996), 137–149. MR 1403648
[25] G.  Strang: Variational crimes in the finite element method. In: The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, A. K. Aziz (ed.), Academic Press, New York, 1972, pp. 689–710. MR 0413554 | Zbl 0264.65068
[26] P.  Sváček: Higher order finite element method for a problem with nonlinear boundary condition. In: Proc. of the 13th Summer School “Software and Algorithms of Numerical Mathematics”, University of West Bohemia in Pilsen, 1999, pp. 301–308.
[27] A. Ženíšek: Nonhomogeneous boundary conditions and curved triangular finite elements. Appl. Math. 26 (1981), 121–141. MR 0612669
[28] A.  Ženíšek: The finite element method for nonlinear elliptic equations with discontinuous coefficients. Numer. Math. 58 (1990), 51–77. DOI 10.1007/BF01385610 | MR 1069653
[29] A.  Ženíšek: Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations. Academic Press, London, 1990. MR 1086876
[30] M.  Zlámal: Curved elements in the finite element method, I. SIAM J.  Numer. Anal. 10 (1973), 229–240. DOI 10.1137/0710022 | MR 0395263
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