Interior estimates for the wavelet Galerkin method
This work is concerned with the use of wavelet bases in determining approximate solutions of partial differential equations. The model problem involves the Poisson equation, which of course is considered in weak or variational form. Solutions are sought in a space \(V_m\) spanned by a wavelet basis. What is of particular interest in this work is the derivation of interior estimates; specifically, suppose that \(\Omega_0\) and \(\Omega_1\) are two subdomains of the domain \(\Omega\) on which the problem is defined, with the property that \(\Omega_0\subset\subset\Omega_1\subset\subset\Omega\). Then for a general class of discretization spaces it has been shown by \textit{J. A. Nitsche} and \textit{A. H. Schatz} [Math. Comput. 28, 937-958 (1974; Zbl 0298.65071)] that the error in the domain \(\Omega_0\) can be bounded according to \[ |u-u_h|_{H^1(\Omega_0)}\leq C(h^{\ell-1}|u|_{H^\ell(\Omega_1)}+|u-u_h|_{H^{-p}(\Omega)})\tag{1} \] provided that \(u\in H^\ell(\Omega_1)\). Thus a high rate of convergence may be obtained on the domain \(\Omega_0\) if \(u\) is regular on the larger subdomain \(\Omega_1\), and provided that an estimate holds in a negative norm for less smooth solutions. The goal of the authors is to show that the estimate (1) holds also for wavelet bases, under suitable assumptions. The framework is that of biorthogonal wavelets [see, for example \textit{A. Cohen, I. Daubechies} and \textit{J.-C. Feauveau}, Commun. Pure Appl. Math. 45, No. 5, 485-560 (1992; Zbl 0776.42020)]. The authors devote a section to reviewing this family of bases, and setting out the assumptions that the bases are required to satisfy: these include assumptions about the support, and inequalities due to Berstein and Jackson. It is then shown that these assumptions hold for classical biorthogonal bases. Eventually, it is shown that an interior error estimate of type (1) holds under various conditions on the integers \(\ell\) and \(p\). In fact the authors are able to derive more general estimates in the \(H^s\)-norm, for integers \(s\) satisfying \(0\leq s\leq\ell\), with the rate of convergence \(\ell-1\) in (1) replaced by \(\ell-s\). The paper concludes with three concrete examples, including one involving the Dirichlet problem with Lagrange multipliers.
scientific article; zbMATH DE number 1096030