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Riesz bases of wavelets and applications to numerical solutions of elliptic equations. (English) Zbl 1248.42031

This interesting paper studies the numerical solutions of the biharmonic equation and general elliptic equations of fourth-order by using spline Riesz bases of wavelets in Sobolev spaces. Using cubic splines on the interval \([0,1]\) and the square \([0,1]^2\) with dyadic partition points, the authors first build a multiresolution analysis \(\{V_n\}_{n=3}^\infty\) and then discuss its approximation property in Section 2. For the variational solution \(u_n\) to the biharmonic equation using the space \(V_n\) in (3.1), the key convergence rates are established in Theorem 3.1, in particular, see formula (3.2). The detailed construction and the analysis of such Riesz bases of wavelets in Sobolev spaces using cubic splines are presented in Sections 4–6. Many convincing numerical simulations for the biharmonic equation and general elliptic equations of fourth-order are provided in Sections 8–10. The proposed wavelet-based method in this paper is shown to be much superior to many other known schemes in the literature, for example, the condition numbers of the stiffness matrices associated with such wavelet bases are relatively small and uniformly bounded.
Reviewer: Bin Han (Edmonton)

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
41A15 Spline approximation
41A25 Rate of convergence, degree of approximation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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