Convergence Analysis of Wavelet Schemes for Convection-Reaction Equations under Minimal Regularity Assumptions.

In this paper, we analyze convergence rates of wavelet schemes for time-dependent convection-reaction equations within the framework of the Eulerian--Lagrangian localized adjoint method (ELLAM). Under certain minimal assumptions that guarantee $ H^1 $-regularity of exact solutions, we show that a generic ELLAM scheme has a convergence rate $ \mathcal{O}(h/\sqrt{\Delta t} + \Delta t) $ in $ L^2 $-norm. Then, applying the theory of operator interpolation, we obtain error estimates for initial data with even lower regularity. Namely, it is shown that the error of such a scheme is $ \mathcal{O}((h/\sqrt{\Delta t})^\theta + (\Delta t)^\theta) $ for initial data in a Besov space $ \displaystyle B^\theta_{2,q} (0 < \theta < 1, 0 < q <= infinity) $. The error estimates are {a priori} and optimal in some cases. Numerical experiments using orthogonal wavelets are presented to illustrate the theoretical estimates. [ABSTRACT FROM AUTHOR]