Wavelet Solution of Plane Elasticity Problem in the Upper Half-plane

The plane elasticity problem includes plane strain problem and plane stress problem which are widely applied in mechanics and engineering. In this article, we first reduce the plane elasticity problem in the upper half-plane into natural boundary integral equation and then apply wavelet-Galerkin method to deal with the numerical solution of the natural boundary integral equation. The test and trial functions used are the scaling basis functions of Shannon wavelet. In our fast algorithm, the computational formulae of entries of the stiffness matrix yield simple close-form and only 3 K entries need to be computed for one 4 K ‐ 4 K stiffness matrix.     

Trigonometric wavelet method for some elliptic boundary value problems

In this paper, we apply trigonometric wavelet method to investigate the numerical solution of elliptic boundary value problem in an exterior unit disk. The simple computation formulae of the entries in the stiffness matrix are obtained. It shows that we only need to compute 2(j²+1) elements of one ²j+2ײj+2 stiffness matrix. Moreover, the error estimates of the approximation solutions are given and some test examples are presented.     

Interior maximum norm estimates for finite element discretizations of the Stokes equations

Interior estimates are proved in the L ^∞ norm for stable finite element discretizations of the Stokes equations on translation invariant meshes. These estimates yield information about the quality of the finite element solution in subdomains a positive distance from the boundary. While they have been established for second-order elliptic problems, these interior, or local, maximum norm estimates for the Stokes equations are new. By applying finite differenciation methods on a translation invariant mesh, we obtain optimal convergence rates in the mesh size h in the maximum norm. These results can be used for analyzing superconvergence in finite element methods for the Stokes equations.     

Finding off‐diagonal entries of the inverse of a large symmetric sparse matrix

The method fast inverse using nested dissection (FIND) was proposed to calculate the diagonal entries of the inverse of a large sparse symmetric matrix. In this paper, we show how the FIND algorithm can be generalized to calculate off‐diagonal entries of the inverse that correspond to ‘short’ geometric distances within the computational mesh of the original matrix. The idea is to extend the downward pass in FIND that eliminates all nodes outside of each node cluster. In our advanced downwards pass, it eliminates all nodes outside of each ‘node cluster pair’ from a subset of all node cluster pairs. The complexity depends on how far (i,j) is from the main diagonal. In the extension of the algorithm, all entries of the inverse that correspond to vertex pairs that are geometrically closer than a predefined length limit l will be calculated. More precisely, let α be the total number of nodes in a two‐dimensional square mesh. We will show that our algorithm can compute O(α^{3 ∕ 2 + 2ε}) entries of the inverse in O(α^{3 ∕ 2 + 2ε}) time where l = O(α^{1 ∕ 4 + ε}) and 0 ≤ ε ≤1 ∕ 4. Numerical examples are given to illustrate the efficiency of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices

We consider an infinite Jacobi matrix with off-diagonal entries dominated by the diagonal entries going to infinity. The corresponding self-adjoint operator J has discrete spectrum and our purpose is to present results on the approximation of eigenvalues of J by eigenvalues of its finite submatrices.     

On Boundary Regularity of the Navier–Stokes Equations

Abstract

We study boundary regularity of weak solutions of the Navier–Stokes equations in the half-space in dimension n ≥ 3. We prove that a weak solution u which is locally in the class L ^{p, q} with 2/p + n/q = 1, q > n near boundary is Hölder continuous up to the boundary. Our main tool is a pointwise estimate for the fundamental solution of the Stokes system, which is of independent interest.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe

The nonsteady Navier–Stokes equations are considered in a thin infinite pipe with the small diameter ? in the case of the Reynolds number of order ?. The time-dependent flow rate is a given function. The complete asymptotic expansion is constructed and justified. The error estimate of order O(? ^J ) for the difference of the exact solution and the J-th asymptotic approximation is proved for any real J.     

Biorthogonal wavelet approximation for the coupling of FEM-BEM

Summary. We apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior Dirichlet boundary value problem for the two dimensional Poisson equation. Adopting biorthogonal wavelet matrix compression to the boundary terms with N degrees of freedom, we show that the resulting compression strategy fits the optimal convergence rate of the coupling Galerkin methods, while the number of nonzero entries in the corresponding stiffness matrices is considerably smaller than . Received December 3, 1999 / Revised version received September 22, 2000 / Published online December 18, 2001     

Matricial formulae for partitions

The exponential of the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the sum of the inverses of the divisors of n is the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the number of partitions of n. A similar result holds for all pairs of sequences satisfying a special mutual recurrence.     

On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations

In this paper, we consider a piecewise linear collocation method for the solution of a pseudo‐differential equation of order r=0, ?1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three‐point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low‐order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low‐order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]³) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N^?(2?r)/2). Note that, in contrast to well‐known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderón–Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd.     

Determinacy of Bounded Complex Perturbations of Jacobi Matrices

We consider complex Jacobi matrices G which can be decomposed in the form G=J+C, where J is a real Jacobi matrix and C is a complex Jacobi matrix whose entries are uniformly bounded. We prove that the determinacy of the operator defined by G is equivalent to that of J. From this we deduce that the determinacy of G is equivalent to the coincidence between the domains of definition of the operators G and its adjoint G*.     

Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω₀ = ℝ^{n –1} × (–1, 1), n ≥ 2, in L^q ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω⁺₀ = ℝ^{n –1} × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–₀ = ℝ^{n –1} × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ²‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of euler-stokes splitting of the navier-stokes equations

We consider approximation by partial time steps of a smooth solution of the Navier-Stokes equations in a smooth domain in two or three space dimensions with no-slip boundary condition. For small k > 0, we alternate the solution for time k of the inviscid Euler equations, with tangential boundary condition, and the solution of the linear Stokes equations for time k, with the no-slip condition imposed. We show that this approximation remains bounded in H^2,p and is accurate to order k in L^p for p > ∞. The principal difficulty is that the initial state for each Stokes step has tangential velocity at the boundary generated during the Euler step, and thus does not satisfy the boundary condition for the Stokes step. The validity of such a fractional step method or splitting is an underlying principle for some computational methods. © 1994 John Wiley & Sons, Inc.     

Wavelet bases of Hermite cubic splines on the interval

In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C¹ and supported on [−1,1]. Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval [0,1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. The wavelet basis is used to solve the Sturm–Liouville equation with the Dirichlet boundary condition. Numerical examples are provided. The computational results demonstrate the advantage of the wavelet basis. Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C40, 41A15, 65L60. Research was supported in part by NSERC Canada under Grants # OGP 121336.     

G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

This paper is concerned with a trigonometric Hermite wavelet Galerkin method for the Fredholm integral equations with weakly singular kernel. The kernel function of this integral equation considered here includes two parts, a weakly singular kernel part and a smooth kernel part. The approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric Hermite wavelet constructed by E. Quak [Trigonometric wavelets for Hermite interpolation, Math. Comp. 65 (1996) 683–722] are developed. The use of trigonometric Hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix. It is shown that we only need to compute and store O(N)$\mathrm{O}(N)$ entries for the weakly singular kernel representation matrix with dimensions N²$N^2$ which can reduce the whole computational cost and storage expense. The computational schemes of the resulting matrix elements are provided for the weakly singular kernel function. Furthermore, the convergence analysis is developed for the trigonometric wavelet method in this paper.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Navier–Stokes Equations and Weighted Convolution Inequalities in Groups

We obtain Gevrey regular mild solutions to the incompressible Navier–Stokes equations in R ⁿ with periodic boundary condition in a subset of the variables. The method is based on an extension of Young's convolution inequality in weighted Lebesgue spaces of measurable functions defined on locally compact abelian groups. This generalizes and provides a unified treatment of the Gevrey regularity result of Foias and Temam in the space periodic case and those of Le Jan and Sznitman and Lemarié–Rieusset in the whole space with no boundary.