Quadratic Spline Wavelets with Short Support Satisfying Homogeneous Boundary Conditions
This work provides a more efficient wavelet basis for solving second-order elliptic problems with Dirichlet boundary conditions, offering computational advantages in terms of iteration count and per-iteration cost.
The paper constructs a new quadratic spline-wavelet basis on the interval and unit square that satisfies homogeneous Dirichlet boundary conditions, achieving the shortest support among known quadratic spline wavelets for such boundary conditions. The resulting stiffness matrices have uniformly bounded and small condition numbers, and numerical examples show that the Galerkin and adaptive wavelet methods using this basis require fewer iterations and fewer floating-point operations per iteration compared to other quadratic spline wavelet bases.
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.