Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems

The numerical approximation by a lower‐order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving semisingular perturbation problems. The quasi‐optimal‐order error estimates are proved in the ε‐weighted H¹‐norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε‐weighted H¹‐norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Convergence of a fourth-order singular perturbation of the n-dimensional radially symmetric Monge–Ampère equation

This paper concerns with the convergence analysis of a fourth-order singular perturbation of the Dirichlet Monge–Ampère problem in the n-dimensional radial symmetric case. A detailed study of the fourth- order problem is presented. In particular, various a priori estimates with explicit dependence on the perturbation parameter ε are derived, and a crucial convexity property is also proved for the solution of the fourth-order problem. Using these estimates and the convexity property, we prove that the solution of the perturbed problem converges uniformly and compactly to the unique convex viscosity solution of the Dirichlet Monge–Ampère problem. Rates of convergence in the H^k-norm for k = 0, 1, 2 are also established.     

Correctors for some asymptotic problems

In the theory of anisotropic singular perturbation boundary value problems, the solution u _ɛ does not converge, in the H ¹-norm on the whole domain, towards some u ₀. In this paper we construct correctors to have good approximations of u _ɛ in the H ¹-norm on the whole domain. Since the anisotropic singular perturbation problems can be connected to the study of the asymptotic behaviour of problems defined in cylindrical domains becoming unbounded in some directions, we transpose our results for such problems.     

Pressure Stabilization Method for Steady Viscous Flows in a System of Pipes

This paper deals with a singular perturbation of the stationary Stokes and Navier-Stokes systems. The term ε²Δp is added to the continuity equation, where ε is a small parameter. For a domain with cylindrical outlets to infinity and exponentially decaying data, existence and uniqueness of solutions under flux conditions at infinity are established for the linear problem and also for the nonlinear problem in the case of small data. Asymptotically exact estimates are proved for ε tending to zero. For sufficiently regular data, these estimates imply the convergence in H _loc ^5/2−δ for the velocity parts and in H _loc ^3/2−δ for the pressure parts, respectively. Bibliography: 17 titles.Dedicated to V. A. Solonnikov on the occasion of his 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 107–133.     

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [−1,1]. It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [−1,1]. Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O(h^α|lnh|) and O(h|lnh|) in the classes of functions H^α([−1,1]) and C¹([−1,1]), respectively.     

Optimal convergence analysis of an immersed interface finite element method

We analyze an immersed interface finite element method based on linear polynomials on noninterface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Optimal error estimates are derived in the broken H ¹-norm and L ²-norm.     

Asymptotic properties of Cramér-Smirnov statistics—A new approach

Let Y₁,…, Y_n be independent identically distributed random variables with distribution function F(x, θ), θ = (θ′₁, θ′₂), where θ_i (i = 1, 2) is a vector of p_i components, p = p₁ + p₂ and for θI, an open interval in ^p, F(x, θ) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under H_n: θ₁ = θ₁₀ + n^−1/2γ, θ₂ unspecified, where γ is a given vector independent of n, is the distribution of a sum of weighted noncentral χ₁² variables whose weights are eigenvalues of a covariance function of a Gaussian process and noncentrality parameters are Fourier coefficients of the mean function of the Gaussian process. Further, the author exploits the special form of the covariance function by using perturbation theory to obtain the noncentrality parameters and the weights. The technique is applicable to other goodness-of-fit statistics such as U² [G. S. Watson, Biometrika 48 (1961), 109–114].     

The Complexity of Indefinite Elliptic Problems with Noisy Data

We study the complexity of second-order indefinite elliptic problems −div(au) +bu=f(with homogeneous Dirichlet boundary conditions) over ad-dimensional domain Ω, the error being measured in theH¹(Ω)-norm. The problem elementsfbelong to the unit ball ofW^{r, p}, (Ω), wherep [2, ∞] andr>d/p. Information consists of (possibly adaptive) noisy evaluations off,a, orb(or their derivatives). The absolute error in each noisy evaluation is at most δ. We find that thenth minimal radius for this problem is proportional ton^−r/d+ δ and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the -complexity (minimal cost of calculating an -approximation) for this problem, said bounds depending on the costc(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation isc(δ) = δ^−s(fors> 0), then the complexity is proportional to (1/)^{d/r + s}.     

Superconvergence in the projected-shear plate-bending finite element method

A projected-shear finite element method for periodic Reissner–Mindlin plate model are analyzed for rectangular meshes. A projection operator is applied to the shear stress term in the bilinear form. Optimal error estimates in the L²-norm, the H¹-norm, and the energy norm for both displacement and rotations are established and gradient superconvergence along the Gauss lines is justified in some weak senses. All the convergence and superconvergence results are uniform with respect to the thickness parameter t. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 367–386, 1998     

Finite element superconvergence approximation for one‐dimensional singularly perturbed problems

Superconvergence approximations of singularly perturbed two‐point boundary value problems of reaction‐diffusion type and convection‐diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of (N^?1 ln (N + 1))^p+1 in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate that the error estimates are sharp; in particular, the logarithmic factor is not removable. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 374–395, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10001     

On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE Hk

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.     

Finite element methods for semilinear parabolic interface problems

In this exposition we study the finite element methods for second-order semilinear parabolic interface problems in two dimensional convex polygonal domains with smooth interface. Both semidiscrete and fully discrete schemes are analyzed. Optimal order error estimates in the L²(0, T; H¹(Ω))-norm are established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global Existence and Convergence Rates of Smooth Solutions for the 3-D Compressible Magnetohydrodynamic Equations Without Heat Conductivity

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H³-norm of the initial perturbation around a constant states is small enough and its L¹-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

Finite volume methods on Voronoi meshes

Two cell-centered finite difference schemes on Voronoi meshes are derived and investigated. Stability and error estimates in a discrete H¹-norm for both symmetric and nonsymmetric problems, including convection dominated, are proven. The theoretical results are illustrated with several numerical experiments. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:193–212, 1998     

Mixed covolume method for parabolic problems on triangular grids

In this paper, we present a mixed covolume method for parabolic equations on triangular grids. This method use the lowest order Raviart–Thomas (R–T) mixed finite element space as the trial space. We prove the optimal order of convergence for the approximate pressure and velocity in L²-norm. Furthermore, we obtain the quasi-optimal error estimates for the approximate pressure in L^∞-norm.     

A finite element method for a noncoercive elliptic problem with Neumann boundary conditions

We consider a noncoercive convection-diffusion problem with Neumann boundary conditions appearing in modeling of magnetic fluid seals. The associated operator has a non-trivial one-dimensional kernel spanned by a positive function. A discretization is proposed preserving these properties. Optimal error estimates in the H¹-norm are based on a discrete stability result. Numerical results confirm the theoretical predictions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An explicit extension formula of bounded holomorphic functions from analytic varieties to strictly convex domains

We consider a strictly convex domain D ⁿ and m holomorphic functions, φ₁,…, φ_m, in a domain . We set V = {z ε Ω: φ₁(z) = ··· = φ_m(z) = 0}, M = V ∩ D and ∂M = V ∩ ∂D. Under the assumptions that the variety V has no singular point on ∂M and that V meets ∂D transversally we construct an explicit kernel K(ζ, z) defined for ζ ε ∂M and z ε D so that the integral operator Ef(z) = ∝ _{ζ ε ∂M} f(ζ) K(ζ, z) (z ε D), defined for f ε H^∞(M) (using the boundary values f(ζ) for a.e. ζ ε ∂M), is an extension operator, i.e., Ef(z) = f(z) for z ε M and furthermore E is a bounded operator from H^∞ to H^∞(D).