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Convergence of fourth order compact difference schemes for three-dimensional convection-diffusion equations. (English) Zbl 1140.65074
The paper presents a high order compact finite difference scheme constructed on a 19-point stencil using Steklov averaging operators, for the Dirichlet boundary valueproblem for three-dimensional convection-diffusion equation with constant coefficients in the unit cube: $$\cases \Delta u+\sum_{\alpha=1}^3 \lambda_\alpha\frac{\partial u}{\partial x_\alpha}=f(x),&x\in\Omega,\ \lambda_\alpha=\text{const}\\ u(x)=0,&x\in\Gamma \endcases.\tag1$$ The first part is an introduction concerning Sobolev spaces and appropiate norms, the finite difference quotients (forward, backward, and central, respectively). The second part focuses on the construction of the derivation of the 19-points compact finite difference schemes for (1). In the third part the authors derive an a priori estimate of the discretization error, of type: $$\| y-u\|_{W_2^m(\omega)}\leq \text{ch}^{s-m}\| u\|_{W_2^s(\Omega)}$$ where $y$ is the solution of the finite difference scheme and $c$ denotes a positive generic constant, independent of $h$ and $u.$ Thus, the real parameter $s$ satisfies $\max(1.5,m)<s\leq m+4,$ $m=0,1,2,$ under the assumption that the solution of the original boundary-value problem (1) belongs to the Sobolev space $W_2^s(\Omega).$ The error estimates are derived using techniques that employ the generalized Bramble-Hilbert lemma. The fourth part concerns the estimate of the convergence rate. The comparisons with other methods and the main conclusions are given in the last part.

##### MSC:
 65N12 Stability and convergence of numerical methods (BVP of PDE) 65N06 Finite difference methods (BVP of PDE) 65N15 Error bounds (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems
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