Multidomain Local Fourier Method for PDEs in Complex Geometries

but with higher frequencies and amplitudes of the oscillating components), only a low accuracy can be achieved after an aaordable number of PCG iterations. The reason is that a strip-wise domain decomposition cannot handle large changes in non-constant coeecients in the lengthwise direction (see explanation at the end of section (5.1)). More general domain decompositions in the form of rectangular boxes will be able to treat eeciently any complicated geometries. where Laplacian r 2 r;; is written in polar coordinates. This equation is solved repeatedly at each time step by the generalized MDLF algorithm as described in previous sections. The strip-wise decomposition Eq.(5.4) of the transformed (rectangular) domain ! is carried out. For typical used in computations, 1 that justiies the implementation of the local matching procedure for the Helmholtz type equation (5.9), was discussed in section 2. As an example we consider the annular region plotted in Fig.7. The inner and outer boundaries are given, respectively, by The source function in Eq.(5.5) is f(x; y) = e x + e y and the boundary values are (x; y) = e x + e y ; (x; y) 2 @. As t ! 1, the solution u(x; y; t) approaches the steady state regime u 1 (x; y) = e x + e y ; (x; y) 2. The maximum relative error in the steady solution is given in Table 3. It decays exponentially fast as the resolution in the azymuthal direction increases.