## 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).