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A Schwarz additive method with high order interface conditions and nonoverlapping subdomains. (English) Zbl 0926.65098
This is an interesting paper showing some new possibilities of the Schwarz method. The additive Schwarz method is applied to the model boundary value problem of the following form: ${u\over{\varepsilon^2}}-\Delta u=f \text{ in }\Omega_d,\;\;u=0 \text{ on } \partial \Omega_d,$ where $$\Omega_d$$ is a rectangle. Since $$\varepsilon^{-2}=\Delta t$$ is the time-step of the time-discretisation of the heat equation, hence $$\Delta t$$ is small. The rectangle $$\Omega_d$$ is divided into nonoverlapping subrectangles. On the interfaces, boundary conditions of special kind are applied. It is shown on a simplified model, where $$\Omega_d={\mathbb R}^2$$ and the interface is the $$y-axis$$, that in this case the best convergence of the method occurs when the interface operator is of the form $${\partial\over {\partial x}}-\Lambda$$, where $$\Lambda$$ is the operator with the symbol $$\lambda(k)=\sqrt{{1\over\varepsilon^2}+k^2}$$. This is in fact the symbol of the original operator $${1\over{\varepsilon^2}}-\Delta$$ after Fourier transform with respect to the variable $$y$$, and $$k$$ is dual to the variable $$y$$. The author suggests to take the second order Taylor series approximation of the symbol $$\lambda(k)$$ at $$k=0$$. This results in the interface operator of the form $${\partial\over{\partial x}}-({1\over\varepsilon}-{\varepsilon\over 2} {\partial^2\over {\partial^2y}})$$. It is shown that for such interface operators on the sides of the subrectangles and some additional interface conditions at the corners, the Schwarz method converges. There is no information about the convergence rate.
MSC:
 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 65F10 Iterative numerical methods for linear systems 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:
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