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 $B_{2,q}^{\theta}(0 < \theta < l, 0 < q \leq \infty)$. The error estimates are a priori and optimal in some cases. Numerical experiments using orthogonal wavelets are presented to illustrate the theoretical estimates.