# On the finite element analysis of problems with nonlinear Newton boundary conditions in nonpolygonal domains

Miloslav Feistauer; Karel Najzar; Veronika Sobotíková

Applications of Mathematics (2001)

- Volume: 46, Issue: 5, page 353-382
- ISSN: 0862-7940

## Access Full Article

top## Abstract

top## How to cite

topFeistauer, Miloslav, Najzar, Karel, and Sobotíková, Veronika. "On the finite element analysis of problems with nonlinear Newton boundary conditions in nonpolygonal domains." Applications of Mathematics 46.5 (2001): 353-382. <http://eudml.org/doc/33092>.

@article{Feistauer2001,

abstract = {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.},

author = {Feistauer, Miloslav, Najzar, Karel, Sobotíková, Veronika},

journal = {Applications of Mathematics},

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; convergence; Poisson's equation; nonlinear boundary conditions; finite elements; variational crimes},

language = {eng},

number = {5},

pages = {353-382},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {On the finite element analysis of problems with nonlinear Newton boundary conditions in nonpolygonal domains},

url = {http://eudml.org/doc/33092},

volume = {46},

year = {2001},

}

TY - JOUR

AU - Feistauer, Miloslav

AU - Najzar, Karel

AU - Sobotíková, Veronika

TI - On the finite element analysis of problems with nonlinear Newton boundary conditions in nonpolygonal domains

JO - Applications of Mathematics

PY - 2001

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 46

IS - 5

SP - 353

EP - 382

AB - 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.

LA - eng

KW - 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; convergence; Poisson's equation; nonlinear boundary conditions; finite elements; variational crimes

UR - http://eudml.org/doc/33092

ER -

## References

top- 10.1016/S0307-904X(81)80024-8, Appl. Math. Modelling 5 (1981), 417–421. (1981) DOI10.1016/S0307-904X(81)80024-8
- Finite element error estimates for nonlinear elliptic equations of monotone type, Numer. Math. 54 (1988), 373–393. (1988) Zbl0643.65058MR0972416
- The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
- 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. (1972) MR0421108
- 10.1007/BF01398378, Numer. Math. 50 (1987), 655–684. (1987) Zbl0646.76085MR0884294DOI10.1007/BF01398378
- Mathematical modelling of an electrolysis process, Comment. Math. Univ. Carolin. 30 (1989), 465–477. (1989) MR1031864
- 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. (1993) MR1318806
- 10.1007/s002110050318, Numer. Math. 78 (1998), 403–425. (1998) MR1603350DOI10.1007/s002110050318
- 10.1080/01630569908816927, Numer. Funct. Anal. Optim. 20 (1999), 835–851. (1999) MR1728186DOI10.1080/01630569908816927
- 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. (2000)
- 10.1051/m2an/1990240404571, RAIRO Modél. Math. Anal. Numér. 24 (1990), 457–500. (1990) MR1070966DOI10.1051/m2an/1990240404571
- 10.1007/BF01396664, Numer. Math. 50 (1987), 451–475. (1987) MR0875168DOI10.1007/BF01396664
- 10.1007/BF01398687, Numer. Math. 52 (1988), 147–163. (1988) MR0923708DOI10.1007/BF01398687
- Monotone operators. A survey directed to applications to differential equations, Appl. Math. 35 (1990), 257–301. (1990) MR1065003
- 10.1137/0731072, SIAM J. Numer. Anal. 31 (1994), 1378–1414. (1994) MR1293521DOI10.1137/0731072
- 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). (1998)
- Nonlinear boundary integral equations for harmonic problems, Applied Mathematics Report AMR98/20, School of Mathematics, The University of New South Wales, Sydney (1998). (1998) MR1738277
- 10.1002/zamm.19940740917, Z. Angew. Math. Mech. 74 (1994), 417–427. (1994) Zbl0823.31006MR1296460DOI10.1002/zamm.19940740917
- Finite element analysis of a nonlinear elliptic problem with a pure radiation condition, In: Applied Nonlinear Analysis, Kluwer, Amsterdam, 1999, pp. 271–280. (1999) MR1727454
- Function Spaces, Academia, Praha, 1977. (1977) MR0482102
- Finite element analysis of a radiation heat transfer problem, J. Comput. Math. 16 (1998), 327–336. (1998)
- An analysis of the hydrodynamics of aluminium reduction cells, J. Electrochem. Soc. 31 (1984), 2251–2259. (1984)
- Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967. (1967) MR0227584
- Finite elements on curved domains, East-West J. Numer. Math. 4 (1996), 137–149. (1996) MR1403648
- 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. (1972) Zbl0264.65068MR0413554
- 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. (1999)
- Nonhomogeneous boundary conditions and curved triangular finite elements, Appl. Math. 26 (1981), 121–141. (1981) MR0612669
- 10.1007/BF01385610, Numer. Math. 58 (1990), 51–77. (1990) MR1069653DOI10.1007/BF01385610
- Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations, Academic Press, London, 1990. (1990) MR1086876
- 10.1137/0710022, SIAM J. Numer. Anal. 10 (1973), 229–240. (1973) MR0395263DOI10.1137/0710022

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.