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 H1-regularity of exact solutions, we show that a generic ELLAM scheme has a convergence rate O(h/ √ Δt +Δt) in L2-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 O((h/ √ Δt)θ + (Δt)θ) for initial data in a Besov space B 2,q(0 < θ < 1, 0 < q ≤ ∞). The error estimates are a priori and optimal in some cases. Numerical experiments using orthogonal wavelets are presented to illustrate the theoretical estimates.