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Keywords:
parabolic convection-diffusion equation; nonlinear Newton boundary condition; Galerkin method; compactness method; finite element approximation; error estimates
Summary:
The paper is concerned with the study of a parabolic initial-boundary value problem with nonlinear Newton boundary condition considered in a two-dimensional domain. The goal is to prove the existence and uniqueness of a weak solution to the problem in the case when the nonlinearity in the Newton boundary condition does not satisfy any monotonicity condition and to analyze the finite element approximation.
References:
[1] Barber S.A.: Soil Nutrient Bioavailability: A Mechanistic Approach. John Wiley & Sons, Inc., New York, 1995.
[2] Bialecki R., Nowak A.J.: Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions. Appl. Math. Model. 5 (1981), 417-421. Zbl 0475.65078
[3] Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York, 1994. MR 1278258 | Zbl 1135.65042
[4] Chow S.S.: Finite element error estimates for nonlinear elliptic equations of monotone type. Numer. Math. 54 (1989), 373-393. MR 0972416
[5] Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. MR 0520174 | Zbl 0547.65072
[6] Ciarlet P.G., Raviart P.A.: 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. MR 0421108
[7] Claassen N., Barber S.A.: Simulation model for nutrient uptake from soil by a growing plant root system. Agronomy Journal 68 (1976), 961-964.
[8] Cwikel M.: Real and complex interpolation and extrapolation of compact operators. Duke Math. J. 65.2 (1992), 333-343. MR 1150590 | Zbl 0787.46062
[9] Dolejší V., Feistauer M., Schwab C.: A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems. preprint, Forschungsinstitut für Mathematik, ETH Zürich, January 2001 (to appear in Calcolo). MR 1901200
[10] Feistauer M.: Mathematical Methods in Fluid Mechanics. The Pitman Monographs and Surveys in Pure and Applied Mathematics 67, Longman Scientific and Technical Series, Harlow, 1993. MR 1266627
[11] Feistauer M., Kalis H., Rokyta M.: Mathematical modelling of an electrolysis process. Comment Math. Univ. Carolinae 30 (1989), 465-477. MR 1031864 | Zbl 0704.35021
[12] Feistauer M., Najzar K.: Finite element approximation of a problem with a nonlinear Newton boundary condition. Numer. Math. 78 (1998), 403-425. MR 1603350 | Zbl 0888.65118
[13] Feistauer M., Najzar K., Sobotíková V.: Error estimates for the finite element solution of elliptic problems with nonlinear Newton boundary conditions. Numer. Funct. Anal. Optim. 20 (1999), 835-851. MR 1728186
[14] Feistauer M., Najzar K., Sobotíková V.: On the finite element analysis of problems with nonlinear Newton boundary conditions in nonpolygonal domains. Appl. Math. 46 (2001), 353-382. MR 1925193 | Zbl 1066.65124
[15] Feistauer M., Najzar K., Sobotíková V., Sváček P.: Numerical analysis of problems with nonlinear Newton boundary conditions. in: Proc. of the 3rd European Conference Numerical Mathematics and Advanced Applications (P. Neittaanmäki, T. Tiihonen, P. Tarvainen, Editors), World Scientific, Singapore, 2000, pp.486-493.
[16] Feistauer M., Sobotíková V.: Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. M$^{2}$AN 24 (1990), 457-500. MR 1070966
[17] Feistauer M., Ženíšek A.: Finite element solution of nonlinear elliptic problems. Numer. Math. 50 (1987), 451-475. MR 0875168
[18] 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. MR 1293521 | Zbl 0815.41008
[19] Ganesh M., Steinbach O.: 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. Zbl 0971.65107
[20] Ganesh M., Steinbach O.: 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 | Zbl 0974.65112
[21] Girault V., Raviart P.A.: Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics 749, Springer-Verlag, Berlin-Heidelberg-New York, 1979. MR 0548867 | Zbl 0441.65081
[22] Křížek M., Liu L., Neittaanmäki P.: 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
[23] Kufner A., John O., Fučík S.: Function Spaces. Academia, Prague, 1977. MR 0482102
[24] Kurzweil J.: Ordinary Differential Equations. Elsevier, Amsterdam-Oxford-New York-Tokyo, 1986. MR 0929466 | Zbl 0756.34003
[25] Lions J.L.: Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Paris, 1969. MR 0259693 | Zbl 0248.35001
[26] Lions J.L., Magenes E.: Problémes aux limites non homogénes et applications. Dunod, Paris, 1968. Zbl 0212.43801
[27] Liu L., Křížek M.: Finite element analysis of a radiation heat transfer problem. J. Comput. Math. 16 (1998), 327-336.
[28] Málek J., Nečas J., Rokyta M., Růžička M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, London, 1996. MR 1409366
[29] Moreau R., Ewans J.W.: An analysis of the hydrodynamics of aluminium reduction cells. J. Electrochem. Soc. 31 (1984), 2251-2259.
[30] Nečas J.: Les méthodes directes en théories des équations elliptiques. Academia, Prague, 1967. MR 0227584
[31] Sváček P.: 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'', West Bohemian University Pilsen, 1999, pp.301-308.
[32] Temam R.: Navier-Stokes Equations. North-Holland, Amsterdam-New York-Oxford, 1977. MR 0603444 | Zbl 1157.35333
[33] Ženíšek A.: Nonhomogeneous boundary conditions and curved triangular finite elements. Appl. Math. 26 (1981), 121-141. MR 0612669
[34] Ženíšek A.: The finite element method for nonlinear elliptic equations with discontinuous coefficients. Numer. Math. 58 (1990), 51-77. MR 1069653
[35] Zlámal M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10 (1973), 229-240. MR 0395263
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