zbMATH — the first resource for mathematics

Finite element approximation for a div-rot system with mixed boundary conditions in non-smooth plane domains. (English) Zbl 0575.65125
The authors examine a finite element method for the numerical approximation of the solution to a div-rot system with mixed boundary conditions in bounded plane domains with piecewise smooth boundary. The solvability of the system both in an infinite and finite dimensional formulation is proved. Piecewise linear element fields with pointwise boundary conditions are used and their approximation properties are studied. Numerical examples indicating the accuracy of the method are given.

65Z05 Applications to the sciences
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
78A25 Electromagnetic theory (general)
Full Text: EuDML
[1] J. H. Bramble A. H. Schatz: Least squares methods for 2m th order elliptic boundary-value problems. Math. Cotp. 25 (1971), 1-32. · Zbl 0216.49202
[2] P. G. Ciarlet: The finite element method for elliptic problems. North-Hiolland Publishing Company, Amsterdam, New York, Oxford, 1978. · Zbl 0383.65058
[3] M. Crouzeix A. Y. Le Roux: Ecoulement d’une fluide irrotationnel. Journées Elements Finis. Université de Rennes, Rennes, 1976.
[4] P. Doktor: On the density of smooth functions in certain subspaces of Sobolev spaces. Comment. Math. Univ. Carolin. 14, 4 (1973), 609-622. · Zbl 0268.46036
[5] G. J. Fix M. D. Gunzburher R. A. Nicolaides: On mixed finite element methods for first order elliptic systems. Numer. Math. 37 (1981), 29-48. · Zbl 0459.65072
[6] V. Girault P. A. Raviart: Finite element approximation of the Navier-Stokes equation. Springer-Verlag, Berlin, Heidelberg, New York, 1979. · Zbl 0413.65081
[7] P. Grisvard: Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. Numerical Solution of Partial Differential Equations III, Academic Press, New York, 1976, 207-274.
[8] J. Haslinger P. Neittaanmäki: On different finite element methods for approximating the gradient of the solution to the Helmholtz equation. Comput. Methods Appl. Mech. Engrg. 42 (1984), 131-148. · Zbl 0574.65123
[9] M. Křížek: Conforming equilibrium finite element methods for some elliptic plane problems. RAIRO Anal. Numer. 17 (1983), 35-65. · Zbl 0541.76003
[10] M. Křížek P. Neittaanmäki: On the validity of Friedrich’s inequalities. Math. Scand.
[11] R. Leis: Anfangsrandwertaufgaben der mathematischen Physik. SFB 74, Bonn, preprint. · Zbl 0474.35002
[12] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. · Zbl 1225.35003
[13] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: an introduction. Elsevier Scientific Publishing Company, Amsterdam, Oxford. New York, 1981.
[14] P. Neittaanmäki R. Picard: On the finite element method for time harmonic acoustic boundary value problems. J. Comput. Math. Appl. 7 (1981), 127-138. · Zbl 0451.76071
[15] P. Neittaanmäki J. Saranen: Finite element approximation of vector fields given by curl and divergence. Math. Meth. Appl. Sci. 3 (1981), 328-335. · Zbl 0456.65069
[16] P. Neittaanmäki J. Saranen: A modified least squares FE-method for ideal fluid flow problems. J. Comput. Appl. Math. 8 (1982), 165-169. · Zbl 0488.76004
[17] J. Saranen: Über die Approximation der Lösungen der Maxwellschen Randwertaufgabe mil der Methode der finiten Elemente. Applicable Anal. 10 (1980), 15 - 30. · Zbl 0454.65079
[18] J. Saranen: A least squares approximation method for first order elliptic systems of plane. Applicable Anal. 14 (1982), 27-42. · Zbl 0478.65065
[19] I. N. Sneddon: Mixed boundary value problems in potential theory. North-Holland Publishing Company, Amsterdam, 1966. · Zbl 0139.28801
[20] J. M. Thomas: Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes. Thesis, Université Paris VI, 1977.
[21] W. L. Wendland E. Stephan G. C. Hsiao: On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Meth. Appl. Sci. 1 (1979), 265-321. · Zbl 0461.65082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.