Riesz bases of wavelets and applications to numerical solutions of elliptic equations

Abstract

Abstract. 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.

Applications of Multigrid Algorithms to Finite Difference Schemes for Elliptic Equations with Variable Coefficients

“….). See [13] for related results. Approximation by quasi-projection operators (see [12]) is a generalization of quasi interpolation introduced in [2].…”

Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions

“…4,5,10,16,[19][20][21][22] In Refs. 2, 3, 12 and 18 cubic spline wavelets on the interval were constructed.…”

“…Spline wavelet or multiwavelet bases where duals are not local are also known. 6,13,15,14,16 The advantage of our construction in comparison with biorthogonal cubic spline wavelets with local duals 2,3,12,18 is that the support of the wavelets is shorter, condition numbers of the corresponding stiffness matrices are smaller and also a simple construction.…”

Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients

“…Another useful tool is the short Haar wavelet transform that was derived and used for solving differential equations in [6][7][8]. Since spline wavelets are known in a closed form and they are smoother and have more vanishing moments than orthogonal wavelets of the same length of support, many wavelet methods using spline wavelets were proposed [9][10][11]. For a review of wavelet methods for solving differential equations, see also [12,13].…”