Quadratic Spline Wavelets with Short Support Satisfying Homogeneous Boundary Conditions

TL;DR

This paper introduces a new quadratic spline-wavelet basis with short support that satisfies boundary conditions, leading to efficient numerical methods with smaller iteration counts and computational costs.

Contribution

The paper presents a novel quadratic spline-wavelet basis with minimal support and boundary condition satisfaction, improving computational efficiency for elliptic problem discretizations.

Findings

Wavelet basis has one vanishing moment and shortest support among similar bases.

Stiffness matrices have uniformly bounded and small condition numbers.

Numerical methods using this basis require fewer iterations and less computation.

Abstract

In the paper, we construct a new quadratic spline-wavelet basis on the interval and a unit square satisfying homogeneous Dirichlet boundary conditions of the first order. Wavelets have one vanishing moment and the shortest support among known quadratic spline wavelets adapted to the same type of boundary conditions. Stiffness matrices arising from a discretization of the second-order elliptic problems using the constructed wavelet basis have uniformly bounded condition numbers and the condition numbers are small. We present quantitative properties of the constructed basis. We provide numerical examples to show that the Galerkin method and the adaptive wavelet method using our wavelet basis requires smaller number of iterations than these methods with other quadratic spline wavelet bases. Moreover, due to the short support of the wavelets one iteration requires smaller number of floating point operations.