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Some error estimates for the box method. (English) Zbl 0634.65105
The paper is devoted to the numerical analysis of the box method (called also box integration method, finite control volume method, or balance method) for solving self-adjoint, positive definite elliptic boundary value problems in plane regions. The analysis is made in terms of the Galerkin procedure known from the finite element method. For the Poisson equation under Dirichlet boundary conditions, the authors derive the estimate \(\|| u-u_ L\|| \leq \|| u-u_ B\|| \leq C\|| u- u_ L\||,\) where \(\||.\||\) denotes the energy norm, and u, \(u_ B\) and \(u_ L\) are the exact solution, the approximate solution generated by the box method and the finite element solution obtained on the primary triangular mesh by means of linear elements, respectively. For the more general boundary value problem \(-div(a\nabla u)+\sigma u=f\) in \(\Omega\) and \(u=0\) on \(\partial \Omega\) with some zero-order term in the differential equation, the authors prove the estimate \(\|| u-u_ L\|| \leq \|| u-u_ B\|| \leq C(\|| u-u_ L\|| +\| u- \bar u_ L\|_{L_ 2(\Omega)}),\) where \(\bar u_ L=\sum u_ L(x_ i){\bar \Phi}_ i(x)\) and \({\bar \Phi}_ i\) denotes the characteristic function of that box to which the vertex \(x_ i\) corresponds.
Reviewer: U.Langer

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
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