A quadratic finite element wavelet Riesz basis

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in $\mathbb{R}^2$. The wavelets are stable in $H^s$ for $|s|<\frac{3}{2}$ and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for $s \in \{-1,0,1\}$ are provided for the unit square.Comment: 13 page

A quadrature algorithm for wavelet Galerkin methods

We consider the wavelet Galerkin method for the solution of boundary integral equations of the first and second kind including integral operators of order r less than zero. This is supposed to be based on an abstract wavelet basis which spans piecewise polynomials of order dT. For example, the bases can be chosen as the basis of tensor product interval wavelets defined over a set of parametrization patches. We define and analyze a quadrature algorithm for the wavelet Galerkin method which utilizes Smolyak quadrature rules of finite order. In particular, we prove that quadrature rules of an order larger than 2dT - r are sufficient to compose a quadrature algorithm for the wavelet Galerkin scheme such that the compressed and quadrature approximated method converges with the maximal order 2dT - r and such that the number of necessary arithmetic operations is less than 풪(N log N) with N the number of degrees of freedom. For the estimates, a degree of smoothness greater or equal to 2[2dT - r]+1 is needed

Element-by-element construction of wavelets satisfying stability and moment conditions

In this paper, we construct a class of locally supported wavelet bases for C&quot;0 Lagrange finite element spaces on possibly non-uniform meshes on n-dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H&quot;3 for vertical stroke s vertical stroke &lt;3/2 (vertical stroke s vertical stroke #&lt;=#1 on Lipschitz' manifolds) and the wavelets can in principal be arranged to have any desired order of vanishing moments. As a consequence, these base can be used e.g. for constructing an optimal solver of discretized H&quot;3-elliptic problems for s in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which for each type of finite element space have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet bases can be implemented efficiently. (orig.)Available from TIB Hannover: RN 8680(145) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Element-By-Element Construction Of Wavelets Satisfying Stability And Moment Conditions

In this paper, we construct a class of locally supported wavelet bases for C 0 Lagrange finite element spaces on possibly non-uniform meshes on n- dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H s for jsj ! 3 2 (jsj 1 on Lipschitz&apos; manifolds), and the wavelets can, in principal, be arranged to have any desired order of vanishing moments. As a consequence, these bases can be used e.g. for constructing an optimal solver of discretized H s -elliptic problems for s in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which, for each type of finite element space, have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet ba..

An operator-customized wavelet-finite element approach for the adaptive solution of second-order partial differential equations on unstructured meshes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Civil and Environmental Engineering, 2005.Includes bibliographical references (p. 139-142).Unlike first-generation wavelets, second-generation wavelets can be constructed on any multi-dimensional unstructured mesh. Instead of limiting ourselves to the choice of primitive wavelets, effectively HB detail functions, we can tailor the wavelets to gain additional qualities. In particular, we propose to customize our wavelets to the problem's operator. For any given linear elliptic second-order PDE, and within a Lagrangian FE space of any given order, we can construct a basis of compactly supported wavelets that are orthogonal to the coarser basis functions with respect to the weak form of the PDE. We expose the connection between the wavelet's vanishing moment properties and the requirements for operator-orthogonality in multiple dimensions. We give examples in which we successfully eliminate all scale-coupling in the problem's multi-resolution stiffness matrix. Consequently, details can be added locally to a coarser solution without having to re-compute the coarser solution.The Finite Element Method (FEM) is a widely popular method for the numerical solution of Partial Differential Equations (PDE), on multi-dimensional unstructured meshes. Lagrangian finite elements, which preserve C⁰ continuity with interpolating piecewise-polynomial shape functions, are a common choice for second-order PDEs. Conventional single-scale methods often have difficulty in efficiently capturing fine-scale behavior (e.g. singularities or transients), without resorting to a prohibitively large number of variables. This can be done more effectively with a multi-scale method, such as the Hierarchical Basis (HB) method. However, the HB FEM generally yields a multi-resolution stiffness matrix that is coupled across scales. We propose a powerful generalization of the Hierarchical Basis: a second-generation wavelet basis, spanning a Lagrangian finite element space of any given polynomial order.by Stefan F. D'Heedene.Ph.D

Parabolic PDEs in Space-Time Formulations: Stability for Petrov-Galerkin Discretizations with B-Splines and Existence of Moments for Problems with Random Coefficients

The topic of this thesis contains influences of numerical mathematics, functional analysis and approximation theory, as well as stochastic respectively uncertainty quantification. The focus is beside the full space-time weak formulation of linear parabolic partial differential equations and their properties, especially the stability of their Petrov-Galerkin approximations. Moreover, the concept of space-time weak formulations is extended to parabolic differential equations with random coefficients, where an additional central aspect of this thesis is the existence of moments of the solution and the optimality of its Petrov-Galerkin solution. In the scope of this work a Matlab code for the numerical realization of the Petrov-Galerkin approximation with B-splines of arbitrary order was developed

Finite element wavelets with improved quantitative properties

In [W. Dahmen, R. Stevenson, Element-by-element construction of wavelets satisfying stability and moment conditions, SIAM J. Numer. Anal. 37 (1) (1999) 319 352 (electronic)], finite element wavelets were constructed on polygonal domains or Lipschitz manifolds that are piecewise parametrized by mappings with constant Jacobian determinants. The wavelets could be arranged to have any desired order of cancellation properties, and they generated stable bases for the Sobolev spaces Hs for jsj <32 (or jsj 1 on manifolds). Unfortunately, it appears that the quantitative properties of these wavelets are rather disappointing. In this paper, we modify the construction from the above-mentioned work to obtain finite element wavelets which are much better conditioned