Adaptive wavelet frame domain decomposition methods for elliptic operator equations

Auteurs

• Manuel Werner

Paramètres

• Manuels
• Manuels universitaires

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This work develops new adaptive numerical wavelet algorithms for solving elliptic operator equations in bounded domains or closed manifolds. To simplify the construction of a wavelet Riesz basis for the solution space, the focus shifts to wavelet frames. Using an overlapping domain decomposition technique, suitable frames can be easily constructed and implemented. The authors demonstrate that classical convergence rates for best N-term approximations with wavelet Riesz bases apply to the wavelet frames considered. An adaptive method based on steepest descent iteration for the frame coordinate representation of the elliptic equation is introduced, along with algorithms utilizing multiplicative and additive Schwarz overlapping domain decomposition methods. The adaptive schemes achieve asymptotically optimal complexity, matching the convergence rate of the best N-term frame approximations. Special numerical quadrature rules for computing the frame representation help keep the overall computational cost proportional to the number of wavelets selected. Numerical tests on non-trivial one- and two-dimensional Poisson and biharmonic model problems validate the theoretical findings, showcasing the efficiency of the domain decomposition approach. Comparisons with a standard adaptive finite element solver reveal that the multiplicative Schwarz method can yield significantly sparser approximations. Additionally, a parallel implementation

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• Année de publication: 2009