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An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. (English) Zbl 0573.65082
This is an interesting paper which gives an inexpensive method to compute the approximate solution of a mixed finite element solution. Consider the model problem \(-div(a\nabla u)=f\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) and assume that a and f are piecewise constants. According to the lowest order mixed method by P. A. Raviart and J. M. Thomas [Lect. Notes Math. 606, 292-315 (1977; Zbl 0362.65089)], u and \(\sigma =a\nabla u\) are approximated by \(\bar u_ h\) and \(\sigma_ h\) unique solution of the problem: find \((\bar u_ h,\sigma_ h)\in C_ h\times R_ h\), such that \[ (1)\quad \int_{\Omega}a^{-1}\sigma_ h\cdot \tau dx+\int_{\Omega}\bar u_ hdiv \tau dx=0,\quad \forall \tau \in R_ h, \] \[ \int_{\Omega}v div \sigma_ hdx=-\int_{\Omega}fv dx,\quad \forall v\in C_ h \] where for any regular triangulation \(\{\tau_ h\}\) of the polygonal domain \(\Omega\), i.e., \(\Omega =\cup_{T\in \{\tau_ h\}}T\), we have set \(C_ h=\{w:w|_ T\) constant}, \(R_ h=\{\vec q:\vec q|_ T\in Q(T)\), \(\forall T\); \(q\cdot n\) continuous at the interelement boundaries} with Q(T) is the restriction to T of span\(\{(1,0),(0,1),(x,y)\}\). Instead to solve (1) which is generally difficult to handle and expensive, the author has proved that \[ (2)\quad \bar u_ h=u_ h(x_ T)-(f/4a)[| x_ T|^ 2- (1/meas(T))\int_{T}| x|^ 2dx]\quad on\quad T, \] and (3) \(\sigma_ h(x)=a\nabla u_ h-(f/2)(x-x_ T)\), \(\forall x\in T\), where \(x_ T\) is the barycenter of the triangle T and where \(u_ h\in V_ h\) is the solution of the ”inexpensive” nonconforming discrete problem \[ (4)\quad \sum_{T}\int_{T}a\nabla u_ h\cdot \nabla v dx=\int_{\Omega}fv dx,\quad \forall v\in V_ h \] with \(V_ h=\{v:v|_ T\in P_ 1(T)\), \(\forall T\); v continuous at the midpoints of the edges; \(v=0\) at midpoints on \(\partial \Omega \}\). Thus, the solution of problem (1) amounts to the solution of problem (4), and next, to the postprocessing (2) and (3).
Reviewer: M.Bernadou

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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