Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes,a new preprocessing approach,called multineighboring grids(MNG),is proposed for solving PDE eigen-problems on an existing grid G(Δ).The linear or multi-linear element,based on box-splines,are taken as the frst stage Kh1Uh=λh1Mh1Uh.In this paper,the j-th stage neighboring-grid scheme is defned asKh jUh=λh j Mh jUh,where Kh j:=Mh j 1Kh1and Mh jUh is to be found as a better mass distribution over the j-th stage neighboring-gridG(Δ),and Kh jcan be seen as an expansion of Kh1on the j-th neighboring-grid with respect to the(j 1)-th mass distribution Mh j 1.It is shown that for an ODE model eigen-problem,the j-th stage scheme with 2j-th order B-spline basis can reach2j-th order accuracy and even(2j+2)-th order accuracy by perturbing the mass matrix.The argument can be extended to high dimensions with separable variable cases.For Laplace eigen-problems with some 2-D and 3-D structured uniform grids,some 2j-th order schemes are presented for j 3.     

Obrechkoff methods having additional parameters for general second-order differential equations

A class of two-step implicit methods involving higher-order derivatives of y for initial value problems of the form y″ = f(t, y, y′) is developed. The methods involve arbitrary parameters p and q, which are determined so that the methods become absolutely stable when applied to the test equation y″ + λy′ + μy = 0. Numerical results for Bessel's and general second-order differential equations are presented to illustrate that the methods are absolutely stable and are of order O(h⁴), O(h⁶) and O(h⁸).     

A highly accurate homogeneous scheme for solving the laplace equation on a rectangular parallelepiped with boundary values in C k, 1

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace??s equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is O(h ⁴), where h is the mesh step, when the boundary functions are from C ^{3, 1}, and the compatibility condition, which results from the Laplace equation, for the second order derivatives on the adjacent faces is satisfied on the edges. Futhermore, it is proved that the order of convergence is O(h ⁶(|lnh| + 1)), when the boundary functions are from C ^{5, 1}, and the compatibility condition for the fourth order derivatives is satisfied. These estimations can be used to justify different versions of domain decomposition methods.     

Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa-tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (▽h ( u-Ihu )1, ▽hvh) h may be estimated as order O ( h2 ) when u ∈ H3 (Ω), where Ihu denotes the bilinear interpolation of u , vh is a polynomial belongs to quasi-Wilson finite element space and ▽h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O ( h2 ) /O ( h3 ) in broken H 1-norm, which is one/two order higher than its interpolation error when u ∈ H3 (Ω) /H4 (Ω). Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O ( h3 ), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.     

On a generalized Minkowski inequality and its relation to dominates for t-norms

Leth:?⁺ → ?⁺ be a continuous strictly increasing function withh(0) = 0. Such functionsh give rise to a generalization of the Minkowski inequality; namely, (1) $$h^{ - 1} (h(a + b) + h(c + d)) \leqq h^{ - 1} (h(a + c) + h(b + d))$$ wherea, b, c, andd are arbitrary non-negative real numbers. Theorem 1 shows that, ifh and logh′ (e ^x ) are both convex functions, thenh satisfies (1). Theorem 2, the major result, demonstrates that, if bothh ₁ andh ₂ satisfy the hypotheses of Theorem 1, then the composition ofh ₁ withh ₂ also satisfies the hypotheses of Theorem 1 and hence the inequality (1). The remainder of the paper shows how (1) and Theorems 1 and 2 impinge on the dominates relation for strict t-norms. In particular, Theorem 3 shows how (1) can be interpreted as equivalent to the dominates relation for two strict t-norms. Theorem 4 shows how to use Theorems 1 and 3 to construct a strict t-norm which dominates a given strict t-norm. And, Theorem 5 shows how Theorem 2 can be used to give a qualified answer of yes to the open question of whether or not the dominates relation is a transitive relation.     

Locking-free nonconforming finite elements for three-dimensional elasticity problem

Two locking-free nonconforming finite elements are presented for three-dimensional elasticity problem with pure displacement boundary condition. Convergence rate of the elements are uniformly optimal with respect to λ. The energy norm and L² norm errors are O(h²) and O(h³), respectively. Lastly, a numerical experiment is carried out, which coincides with the theoretical analysis.     

On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients

Implicit difference schemes of O(k⁴ + k²h² + h⁴), where k0, h 0 are grid sizes in time and space coordinates respectively, are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and Dirichlet boundary conditions. The proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry. The numerical examples are given to illustrate the fourth order convergence of the methods.     

Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients

In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x ₀ for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h ^p+1 |ln h|²) and O(h ^p+2 |ln h|²) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.     

Asymptotic coefficients for Gaussian radial basis function interpolants

Gaussian radial basis functions (RBFs) on an infinite interval with uniform grid pacing h are defined by ?(x;α,h)≡exp(-[α²/h²]x²). The only significant numerical parameter is α, the inverse width of the RBF functions relative to h. In the limit α→0, we demonstrate that the coefficients of the interpolant of a typical function f(x) grow proportionally to exp(π²/[4α²]). However, we also show that the approximation to the constant f(x)≡1 is a Jacobian theta function whose coefficients do not blow up as α→0. The subtle interplay between the complex-plane singularities of f(x) (the function being approximated) and the RBF inverse width parameter α are analyzed. For α≈1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(10⁴) and the error saturation is smaller than machine epsilon, so this α is the center of a “safe operating range” for Gaussian RBFs.     

Mirror property for nonsingular mixed configurations of lines and points in ℝ3

LetM(h, k) denote the topological space whose points are the sets composed ofh lines andk points of ?³ in general position. We say thatf∈M(h, k) is mirror if a path exists inM(h, k) that joinsf and its mirror image after reflection in any plane of ?³. In this paper we are principally concerned about the following problem (an affine version of a problem proposed by Viro for the projective space): Givenh, k≥0, does somef∈M(h, k) exist such thatf is mirror? This question is solved for all cases except whenh≡1 (mod 4) withh≥5 andk=2 or 3.     

Efficient three-level scheme for parabolic equations in cylindrical coordinates in a region with a small hole

An efficient three-level scheme for parabolic equations in cylindrical coordinates is constructed in a region with a small hole. No axial symmetry is assumed. The convergence rate of the scheme is estimated under minimum requirements on the initial data. The estimates are uniform with respect to a small parameter—the inner diameter of the region. The order of convergence is τ + h ², τ^1/2 + h, τ + h, depending on the smoothness of the data.     

Necessary and sufficient consistency conditions for a recursive kernel regression estimate

A recursive kernel estimate ∑_{i = 1}ⁿ Y_iK⧸(x − X_i)h_i)⧸∑_{j = 1}ⁿ K((x − X_j)⧸h_j) of a regression m(x) = E{Y|X = x} calculated from independent observations (X₁, Y₁),…, (X_n, Y_n) of a pair (X, Y) of random variables is examined. ForE|Y|^{1 + δ} < ∞, δ > 0, the estimate is weakly pointwise consistent for almost all (μ) x ∈ R^d, μ is the probability measure of X, if and only if∑_i−1ⁿ h_i^d I_{hi > ɛ } ⧸ ∑_{j = 1}ⁿ h_j^d → 0 as n → ∞, all ɛ > 0, and∑_{i = 1}^∞ h_i^d = ∞, d is the dimension of X. For E|Y|^{1 + δ} < ∞, δ > 0, the estimate is strongly pointwise consistent for almost all (μ) x ∈ R^d, if and only if the same conditions hold. ForE|Y|^{1 + δ} < ∞, δ > 0, weak and strong consistency are equivalent. Similar results are given for complete convergence.     

EQ 1 rot nonconforming finite element method for nonlinear dual phase lagging heat conduction equations

EQ rot 1 nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2 ) one order higher than its interpolation error O(h), the superclose results of order O(h2 ) in broken H1 -norm are obtained. At the same time, the global superconvergence in broken H1 -norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4 ) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQ rot 1 element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

An infinite family of symmetric designs

In this paper, using the construction method of [3], we show that if q>2 is a prime power such that there exists an affine plane of order q?1, then there exists a strongly divisible 2?(q?1)(q^h?1), q^h?1(q?1), q^h?1) design for every h?2. We show that these quasi-residual designs are embeddable, and hence establish the existence of an infinite family of symmetric 2?(q^h+1?q+1,q^h, q^h?1) designs. This construction may be regarded as a generalisation of the construction of [1, Chapter 4, Section 1] and [4].     

Finite element methods for symmetric hyperbolic equations

A finite difference method for the solution of symmetric positive differential equations has already been developped (Katsanis [4]). The finite difference solutions where shown to converge at the rateO(ith ^1/2) ash approaches zero,h being the maximum distance between two adjacent mesh points. Here we try to get a better rate of convergence, using a Rayleigh Ritz Galerkin method. We first give a “weak” formulation of the equations, slightly different from the usual one (Friedrichs [3]), in order to take into account the boundary conditions. We define a finite dimensional subspaceV _h ofH ¹(Ω), in which we look for an approximate solutionu _h . We show that when the exact solutionu is smooth enough, we get the error estimate: $$\left| {u - u_h } \right|L^2 (\Omega ) \leqq C\mathop {\inf }\limits_{v_h \in V_h } \left\{ {\left\| {u - v_h } \right\|H^1 (\Omega ) + \mathop {\sup }\limits_{w_h \in V_h } \frac{{\int\limits_\Gamma {\left| {u - v_h } \right|\left| {w_h } \right|d\Gamma } }}{{\left| {w_h } \right|L^2 (\Omega )}}} \right\}$$ where |·| denotes the Euclidean norm inR ^P . Thus, as is the case for elliptic or parabolic equations, the problem of estimating the error is reduced to questions in approximation theory. When those results are applied to finite element methods, with polynomial approximations of degree ≦k over eachn-simplex we obtain a rate of convergence ofO(h ^k) ash approaches zero,h being the supremum of the diameters of then-simplices.     

A locking-free nonconforming triangular element for planar elasticity with pure traction boundary condition

A new nonconforming triangular element for the equations of planar linear elasticity with pure traction boundary conditions is considered. By virtue of construction of the element, the discrete version of Korn’s second inequality is directly proved to be valid. Convergence rate of the finite element methods is uniformly optimal with respect to λ. Error estimates in the energy norm and L²-norm are O(h²) and O(h³), respectively.     

Higher order numerical methods for solving fractional differential equations

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0<α<1. The order of convergence of the numerical method is O(h ^3?α ). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α>0. The order of convergence of the numerical method is O(h ³) for α≥1 and O(h ^1+2α ) for 0<α≤1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.     

Conjugate connections and Radon's theorem in affine differential geometry

For a given nondegenerate hypersurfaceM ⁿ in affine space ? ⁿ⁺¹ there exist an affine connection ?, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=?h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ? and a nondegenerate metrich on a differentiable manifoldM ⁿ such that ?h is totally symmetric and satisfies the apolarity condition relative toh, canM ⁿ be locally immersed in ? ⁿ⁺¹ in such a way that (?,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ?. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (?,h) is that the conjugate connection of ? relative toh is projectively flat.     

Boundary Multifractal Behaviour for Harmonic Functions in the Ball

It is well known that if h is a nonnegative harmonic function in the ball of $\mathbb R^{d+1}$ or if h is harmonic in the ball with integrable boundary values, then the radial limit of h exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any β?∈?[0,d], the Hausdorff dimension of the set of points ξ on the sphere such that h(rξ) looks like (1???r)^???β is equal to d???β.