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Maxwell equations; finite element method; div-rot system; mixed boundary conditions; piecewise smooth boundary; Piecewise linear element fields; numerical examples
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.
[1] J. H. Bramble A. H. Schatz: Least squares methods for 2m th order elliptic boundary-value problems. Math. Сотр. 25 (1971), 1-32. MR 0295591
[2] P. G. Ciarlet: The finite element method for elliptic problems. North-Hiolland Publishing Company, Amsterdam, New York, Oxford, 1978. MR 0520174 | 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. MR 0336317
[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. DOI 10.1007/BF01396185 | MR 0615890
[6] V. Girault P. A. Raviart: Finite element approximation of the Navier-Stokes equation. Springer-Verlag, Berlin, Heidelberg, New York, 1979. MR 0548867
[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. MR 0466912
[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. DOI 10.1016/0045-7825(84)90022-7 | MR 0737949
[9] M. Křížek: Conforming equilibrium finite element methods for some elliptic plane problems. RAIRO Anal. Numer. 17 (1983), 35-65. DOI 10.1051/m2an/1983170100351 | MR 0695451
[10] M. Křížek P. Neittaanmäki: On the validity of Friedrich's inequalities. Math. Scand. (to appear). MR 0753060
[11] R. Leis: Anfangsrandwertaufgaben der mathematischen Physik. SFB 74, Bonn, preprint. MR 1290369 | Zbl 0474.35002
[12] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. MR 0227584
[13] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: an introduction. Elsevier Scientific Publishing Company, Amsterdam, Oxford. New York, 1981. MR 0600655
[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. DOI 10.1016/0898-1221(81)90111-5 | MR 0619754
[15] P. Neittaanmäki J. Saranen: Finite element approximation of vector fields given by curl and divergence. Math. Meth. Appl. Sci. 3 (1981), 328-335. DOI 10.1002/mma.1670030124 | MR 0657301
[16] P. Neittaanmäki J. Saranen: A modified least squares FE-method for ideal fluid flow problems. J. Comput. Appl. Math. 8 (1982), 165-169. DOI 10.1016/0771-050X(82)90038-9
[17] J. Saranen: Über die Approximation der Lösungen der Maxwellschen Randwertaufgabe mil der Methode der finiten Elemente. Applicable Anal. 10 (1980), 15 - 30. MR 0572804
[18] J. Saranen: A least squares approximation method for first order elliptic systems of plane. Applicable Anal. 14 (1982), 27-42. DOI 10.1080/00036818208839407 | MR 0678492 | Zbl 0478.65065
[19] I. N. Sneddon: Mixed boundary value problems in potential theory. North-Holland Publishing Company, Amsterdam, 1966. MR 0216018 | 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. DOI 10.1002/mma.1670010302 | MR 0548943
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