# Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type

Liping Liu; Michal Křížek; Pekka Neittaanmäki

Applications of Mathematics (1996)

- Volume: 41, Issue: 6, page 467-478
- ISSN: 0862-7940

## Access Full Article

top## Abstract

top## How to cite

topLiu, Liping, Křížek, Michal, and Neittaanmäki, Pekka. "Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type." Applications of Mathematics 41.6 (1996): 467-478. <http://eudml.org/doc/32962>.

@article{Liu1996,

abstract = {A nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions is examined. The problem describes for instance a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree $k\ge 1$ we prove the optimal rates of convergence $\mathcal \{O\}(h^k)$ in the $H^1$-norm and $\mathcal \{O\}(h^\{k+1\})$ in the $L^2$-norm provided the true solution is sufficiently smooth. Considerations are restricted to domains with polyhedral boundaries. Numerical integration is not taken into account.},

author = {Liu, Liping, Křížek, Michal, Neittaanmäki, Pekka},

journal = {Applications of Mathematics},

keywords = {nonlinear boundary value problem; finite elements; rate of convergence; anisotropic heat conduction; nonlinear boundary value problem; finite elements; rate of convergence; anisotropic heat conduction},

language = {eng},

number = {6},

pages = {467-478},

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

title = {Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type},

url = {http://eudml.org/doc/32962},

volume = {41},

year = {1996},

}

TY - JOUR

AU - Liu, Liping

AU - Křížek, Michal

AU - Neittaanmäki, Pekka

TI - Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type

JO - Applications of Mathematics

PY - 1996

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

VL - 41

IS - 6

SP - 467

EP - 478

AB - A nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions is examined. The problem describes for instance a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree $k\ge 1$ we prove the optimal rates of convergence $\mathcal {O}(h^k)$ in the $H^1$-norm and $\mathcal {O}(h^{k+1})$ in the $L^2$-norm provided the true solution is sufficiently smooth. Considerations are restricted to domains with polyhedral boundaries. Numerical integration is not taken into account.

LA - eng

KW - nonlinear boundary value problem; finite elements; rate of convergence; anisotropic heat conduction; nonlinear boundary value problem; finite elements; rate of convergence; anisotropic heat conduction

UR - http://eudml.org/doc/32962

ER -

## References

top- Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris Ser. I Math. 315 (1992), 1159–1164. (1992) MR1194509
- On the existence, uniqueness and convergence of nonlinear mixed finite element methods, Mat. Apl. Comput. 8 (1989), 241–258. (1989) Zbl0709.65080MR1067288
- The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
- A Galerkin method for a nonlinear Dirichlet problem, Math. Comp. 29 (1975), 689–696. (1975) MR0431747
- Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form, Arch. Rational Mech. Anal. 42 (1971), 157–168. (1971) MR0393829
- 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
- Finite element approximation of nonlinear elliptic problems with discontinuous coefficients, RAIRO Modèl. Math. Anal. Numér. 24 (1990), 457–500. (1990) MR1070966
- Compactness method in finite element theory of nonlinear elliptic problems, Numer. Math. 52 (1988), 147–163. (1988) MR0923708
- Weakly continuous operators. Applications to differential equations, Appl. Math. 39 (1994), 45–56. (1994) MR1254746
- Asymptotic ${L}^{\infty}$-error estimates for linear finite element approximations of quasilinear boundary value problems, SIAM J. Numer. Anal. 15 (1978), 418–431. (1978) MR0502037
- Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1977. (1977) MR0473443
- Reliable solution of a quasilinear nonpotential elliptic problem of a nonmonotone type with respect to the uncertainty in coefficients, accepted by J. Math. Anal. Appl. MR1464890
- On a nonpotential and nonmonotone second order elliptic problem with mixed boundary conditions, Stability Appl. Anal. Contin. Media 3 (1993), 85–97. (1993)
- On Galerkin approximations of quasilinear nonpotential elliptic problem of a nonmonotone type, J.Math. Anal. Appl. 184 (1994), 168–189. (1994) MR1275952
- On diagonal dominance of stiffness matrices in 3D, East-West J. Numer. Math. 3 (1995), 59–69. (1995) MR1331484
- On a comparison principle for a quasilinear elliplic boundary value problem of a nonmonotone type, Applicationes Mathematicae 24 (1996), 97–107. (1996) MR1404987
- Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications, Kluwer, Dordrecht, 1996. (1996) MR1431889
- 3d solution of temperature fields in magnetic circuits of large transformers (in Czech), Elektrotechn. obzor 76 (1987), 646–652. (1987)
- Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp. 44 (1985), 303–320. (1985) Zbl0567.65079MR0777266
- Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague, 1967. (1967) MR0227584
- Introduction to the Theory of Nonlinear Elliptic Equations, Teubner, Leipzig, 1983. (1983) MR0731261
- On ${L}_{\infty}$-convergence of finite element approximations to the solution of nonlinear boundary value problem, in: Proc. of Numer. Anal. Conf. (ed. J. H. Miller), Academic Press, New York, 1977, 317–325. MR0513215
- Introduzione al Metodo Degli Elementi Finiti, Lecture Notes, Trento Univ., 1985. (1985)
- Temperature distribution in large transformer cores, Proc. of GANZ Conf. (ed. M. Franyó), Budapest, 1985, 254–261.
- Functional Analysis, Springer-Verlag, Berlin, 1965. (1965) Zbl0126.11504
- Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations, Academic Press, London, 1990. (1990) MR1086876
- The finite element method for nonlinear elliptic equations with discontinuous coeffcients, Numer. Math. 58 (1990), 51–77. (1990)

## 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.