Adaptive Wavelet Methods for Saddle Point Problems---Optimal Convergence Rates

• Authors: Stephan Dahlke;Wolfgang Dahmen;Karsten Urban
• Venue: SIAM Journal on Numerical Analysis
• Year: 2002

In this paper an adaptive wavelet scheme for saddle point problems is developed and analyzed. Under the assumption that the underlying continuous problem satisfies the inf-sup condition, it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity. This means that within a certain range the convergence rate which relates the achieved accuracy to the number of involved degrees of freedom is asymptotically the same as the error of the best wavelet N-term approximation of the solution with respect to the relevant norms. Moreover, the computational work needed to compute the approximate solution stays proportional to the number of degrees of freedom. It is remarkable that compatibility constraints on the trial spaces such as the Ladyzhenskaya--Babuska--Brezzi (LBB) condition do not arise. In the second part the general results are applied to the Stokes problem. Aside from the verification of those requirements on the algorithmic ingredients the theoretical analysis had been based upon, the regularity of the solutions in certain Besov scales is analyzed. These results reveal under which circumstances the work/accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements. This in turn is used to select and interpret some first numerical experiments that are to quantitatively complement the theoretical results for the Stokes problem.