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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 $-\text{div}\left(a\text{grad}\phantom{\rule{4.pt}{0ex}}u\right)=f$ in $G\subset {\Re }^{3}$, $u=g$ on $\partial G$. By substituting $q=-a\text{grad}\phantom{\rule{4.pt}{0ex}}u$ the authors use a weak formulation of the equivalent first order system $q+a\text{grad}\phantom{\rule{4.pt}{0ex}}u=0$, $\text{div}\phantom{\rule{4.pt}{0ex}}q=f$, 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 $O\left({h}^{j}\right)$ to $O\left({h}^{j+2}\right)$ 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 elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65F10 Iterative methods for linear systems 65N15 Error bounds (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems
##### References:
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