Rong-Qing Jia

We investigate Riesz bases of wavelets in Sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth-order. First, we study bicubic splines on the unit square with homogeneous boundary conditions. The approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation. Second, we develop a fairly general theory for Riesz bases of Hilbert spaces equipped with induced norms. Under the guidance of the general theory, we are able to construct wavelet bases for Sobolev spaces on the unit square. The condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded. Third, we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient. Finally, we extend our study to general elliptic equations of fourth-order and demonstrate that our numerical schemes also have superb performance in the general case.

In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in C 1 C^1 wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of C 1 C^1 quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct C 1 C^1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Hölder smoothness exponent of its basis function cannot exceed log 3 ⁡ 11 ( ≈ 2.18266 ) \log _3 11 (\approx 2.18266) , where the critical Hölder smoothness exponent of a function f : R 2 ↦ R f : \mathbb {R}^2\mapsto \mathbb {R} is defined to be \[ ν ∞ ( f ) ≔ sup { ν : f ∈ Lip ⁡ ν } . \nu _\infty (f) \coloneq \sup \{ \nu : f\in \operatorname {Lip} \nu \}. \] On the other hand, for both regular triangular and quadrilateral meshes, we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical Hölder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound log 3 ⁡ 11 \log _3 11 . Consequently, we obtain optimal smoothest C 2 C^2 interpolatory ternary subdivision schemes with two-ring stencils for the regular triangular and quadrilateral meshes. Our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the ℓ p \ell _p -norm joint spectral radius.