Solution of 1D and 2D Poisson's Equation by Using Wavelet Scaling Functions (original) (raw)

Direct Solution of Differential Equations Using a Wavelet-Based Multiresolution Method

2011

The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs). Therefore, the use of wavelets as basis functions in computational analysis holds some promise due to their compact support, orthogonality, localization and multiresolution properties, especially for problems with local high gradient, which would require a dense mesh in tradicional methods, like the FEM. Another possible advantage is the fact that the calculation of the integrals of the inner products of wavelet basis functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of approximating the integral by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness, mass and geometry matrices. In this work, the Galerkin Method has been adapted for the direct solution of diff...

The numerical performance of wavelets for PDEs: the multi-scale finite element

Computational Mechanics, 2000

The research summarized in this paper is part of a multi-year effort focused on evaluating the viability of wavelet bases for the solution of partial differential equations. The primary objective for this work has been to establish a foundation for hierarchical/wavelet simulation methods based upon numerical performance, computational ef®ciency, and the ability to exploit the hierarchical adaptive nature of wavelets. This work has demonstrated that hierarchical bases can be effective for problems with a dominant elliptic character. However, the strict enforcement of orthogonality in the usual L 2 sense is less desirable than orthogonality in the energy norm. This conclusion has led to the development of a multi-scale linear ®nite element based on a hierarchical change-of-basis. This work considers the numerical and computational performance of the hierarchical Schauder basis in a Galerkin context. A unique row-column lumping procedure is developed with multi-scale solution strategies for 1-D and 2-D elliptic partial differential equations.

Direct Solution of Differential Equations Using the Wavelet-Galerkin Method

2010

The use of compactly supported wavelet functions has become increasingly popular in the development of numerical solutions for differential equations, especially for problems with local high gradient. Daubechies wavelets have been successfully used as base functions in several schemes like the Wavelet-Galerkin Method, due to their compact support, orthogonality, and multi-resolution properties. Another advantage of wavelet-based methods is the fact that the calculation of the inner products of wavelet basis functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of approximating the integral by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness, mass and geometry matrices. In this work, the Wavelet-Galerkin Method has been adapted for the direct solution of differential equations in a meshless formulation. This approach enables the use of a mu...

Combining multigrid and wavelet ideas to construct more efficient multiscale algorithms for the solution of Poisson's equation

Eprint Arxiv Physics 0209040, 2002

It is shown how various ideas that are well established for the solution of Poisson's equation using plane wave and multigrid methods can be combined with wavelet concepts. The combination of wavelet concepts and multigrid techniques turns out to be particularly fruitful. We propose a modified multigrid V cycle scheme that is not only much simpler, but also more efficient than the standard V cycle. Whereas in the traditional V cycle the residue is passed to the coarser grid levels, this new scheme does not require the calculation of a residue. Instead it works with copies of the charge density on the different grid levels that were obtained from the underlying charge density on the finest grid by wavelet transformations. This scheme is not limited to the pure wavelet setting, where it is faster than the preconditioned conjugate gradient method, but equally well applicable for finite difference discretizations.

Analysis of Beams and Thin Plates Using the Wavelet-Galerkin Method

International Journal of Engineering and Technology, 2015

The use of wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs), especially for problems with local high gradient. In this work, the Galerkin Method has been adapted for the direct solution of differential equations in a meshless formulation using Daubechies wavelets and Deslauriers-Dubuc interpolating functions (Interpolets). This approach takes advantage of wavelet properties like compact support, orthogonality and exact polynomial representation, which allow the use of a multiresolution analysis. Several examples based on typical differential equations for beams and thin plates were studied successfully.

Finite Elements Based on Deslauriers-Dubuc Wavelets for Wave Propagation Problems

Applied Mathematics, 2016

This paper presents the formulation of finite elements based on Deslauriers-Dubuc interpolating scaling functions, also known as Interpolets, for their use in wave propagation modeling. Unlike other wavelet families like Daubechies, Interpolets possess rational filter coefficients, are smooth, symmetric and therefore more suitable for use in numerical methods. Expressions for stiffness and mass matrices are developed based on connection coefficients, which are inner products of basis functions and their derivatives. An example in 1-D was formulated using Central Difference and Newmark schemes for time differentiation. Encouraging results were obtained even for large time steps. Results obtained in 2-D are compared with the standard Finite Difference Method for validation.

The solution of multi-scale partial differential equations using wavelets

1998

Wavelets are a powerful new mathematical tool which offers the possibility to treat in a natural way quantities characterized by several length scales. In this article we will show how wavelets can be used to solve partial differential equations which exhibit widely varying length scales and which are therefore hardly accessible by other numerical methods. As a benchmark calculation we solve Poisson's equation for a 3-dimensional Uranium dimer. The length scales of the charge distribution vary by 4 orders of magnitude in this case. Using lifted interpolating wavelets the number of iterations is independent of the maximal resolution and the computational effort therefore scales strictly linearly with respect to the size of the system.

THE CONSTRUCTION OF PLATE FINITE ELEMENTS USING WAVELET BASIS FUNCTIONS

In the last years, applying wavelets analysis has called the attention in a wide variety of practical problems, in particular for the numerical solutions of partial differential equations using different methods, as finite differences, semi-discrete techniques or the finite element method. In the construction of wavelet-based elements, instead of traditional polynomial interpolation, scaling and wavelet functions have been adopted to form the shape function to construct elements. Due to their properties, wavelets are very useful when it is necessary to approximate efficiently the solution on non-regular zones. Furthermore, in some cases it is convenient to use the Daubechies wavelet, which has properties of orthogonality and minimum compact support, and provides guaranty of convergence and accuracy of the approximation in a wide variety of situations. The aim of this research is to explore the Galerkin method using wavelets to solve plate bending problems. Some numerical examples, with B-splines and Daubechies, are presented and show the feasibility of our proposal.