## Mixed finite elements for second order elliptic problems in three variables.(English)Zbl 0631.65107

The authors introduce two families of spaces of mixed finite elements to approximate the solution of Dirichlet problems of the form $$-\text{div}(a \text{grad }u) = f$$ in $$G \subset {\mathfrak R}^3$$, $$u = g$$ on $$\partial G$$. By substituting $$q= -a \text{grad }u$$ the authors use a weak formulation of the equivalent first order system $$q + a \text{grad }u = 0$$, $$\text{div }q = f$$, for $$q$$ and $$u$$ to approximate the solution. The first family of spaces are spaces over simplices with flat faces in G; boundary simplices have one curved face lying in the boundary of $$G$$. The second family are spaces over cubes (i.e. rectangular parallelepipeds) in $$G$$ and simplicial boundary elements with one curved face as in the first family. The elements are based on polynomials of total degree $$j$$ for the vector variable $$q$$ and total degree $$j-1$$ for the scalar variable u. Error estimates in $$L^2$$ and $$H^{-s}$$ are derived. In addition it is shown that the solution of the resulting algebraic equations may be simplified by introducing a Lagrange multiplier to enforce the continuity of the normal components of the approximation of q across interelement boundaries; this method allows a post-processing of the approximation of u which improves the convergence from $$O(h^j)$$ to $$O(h^{j + 2})$$ for $$j > 1$$. Finally, an Arrow-Hurwitz-type alternating-direction technique for the solution of the algebraic equations is described briefly.
Reviewer: J. Weisel

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65N15 Error bounds for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
Full Text:

### References:

 [1] Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing, and error estimates. M2AN,19, 7-32 (1985) · Zbl 0567.65078 [2] Arnold, D.N., Brezzi, F., Douglas, J., Jr.: Peers: a new mixed finite element for plane elasticity. Japan J. Appl. Math.1, 347-367 (1984) · Zbl 0633.73074 [3] Brezzi, F., Douglas, J., Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.47, 217-235 (1985) · Zbl 0599.65072 [4] Brezzi, F., Douglas, J., Jr., Marini, L.D.: Variable degree mixed methods for secod order elliptic problems. Mat. Apl. Comput.4, 19-34 (1985) · Zbl 0592.65073 [5] Brown, D.C.: Alternating-direction iterative schemes for mixed finite element methods for second order elliptic problems. Thesis, University of Chicago 1982 [6] Douglas, J., Jr.: Alternating direction methods for three space variables. Numer. Math.4, 41-63 (1962) · Zbl 0104.35001 [7] Douglas, J., Jr., Durán, R., Pietra, P.: Formulation of alternating-direction iterative methods for mixed methods in three space. Proceedings of the Simposium Internacional de Analisis Numérico, Madrid, September 1985 [8] Douglas, J., Jr., Durán, R., Pietra, P.: Alternating-direction iteration for mixed finite element methods. Proceedings of the Seventh International Conference on Computing Methods in Applied Sciences and Engineering, Versailles, December 1985 [9] Douglas, J., Jr., Pietra, P.: A description of some alternating-direction iterative techniques for mixed finite element methods. Proceedings. SIAM/SEG/SPE conference, Houston, January 1985 [10] Douglas, J., Jr., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comput.44, 39-52 (1985) · Zbl 0624.65109 [11] Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev space. Math. Comput.34, 441-463 (1980) · Zbl 0423.65009 [12] Fraeijs de Veubeke, B.X.: Displacement and equilibrium models in the finite element method. In: Stress analysis (O.C. Zienkiewicz, G. Holister, eds.). New York: John Wiley 1965 · Zbl 0359.73007 [13] Fraeijs de Veubeke, B.X.: Stress function approach. World Congress on the Finite Element Method in Structural Mechanics. Bournemouth, 1965 [14] Girault, V., Raviart, P.A.: Finite element approximation of the Navier-Stokes equation. Lecture Notes in Mathematics, Vol. 749. Berlin, Heidelberg, New York: Springer 1979 · Zbl 0413.65081 [15] Nedelec, J.C.: Mixed finite elements inR 3. Numer. Math.35, 315-341 (1980) · Zbl 0419.65069 [16] Nedelec, J.C.: A new family of mixed finite elements inR 3 Numer. Math.50, 57-82 (1986) · Zbl 0625.65107 [17] Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg36, 9-15 (1970/1971) · Zbl 0229.65079 [18] Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. Mathematical aspects of the finite element method. Lecture Notes in Mathematics, Vol. 606. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0362.65089
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.