Refined Error Estimates for Radial Basis Function Interpolation

We discuss new and refined error estimates for radial-function scattered-data interpolants and their derivatives. These estimates hold on R ^d , the d-torus, and the 2-sphere. We employ a new technique, involving norming sets, that enables us to obtain error estimates, which in many cases give bounds orders of magnitude smaller than those previously known.     

Error estimation and adaptive discretization for the discrete stochastic Hamilton–Jacobi–Bellman equation

Summary. Generalizing an idea from deterministic optimal control, we construct a posteriori error estimates for the spatial discretization error of the stochastic dynamic programming method based on a discrete Hamilton–Jacobi–Bellman equation. These error estimates are shown to be efficient and reliable, furthermore, a priori bounds on the estimates depending on the regularity of the approximate solution are derived. Based on these error estimates we propose an adaptive space discretization scheme whose performance is illustrated by two numerical examples.Mathematics Subject Classification (2000): 93E20, 65N50, 49L20, 49M25, 65N15Acknowledgments. This research was supported by the Center for Empirical Macroeconomics, University of Bielefeld. The support is gratefully acknowledged. I would also like to thank an anonymous referee who suggested several improvements for the paper.     

Finite element approximation of a semilinear elliptic problem with a singular nonlinearity

Given , we consider the following problem: find , such that where or 3, and in . We prove and error bounds for the standard continuous piecewise linear Galerkin finite element approximation with a (weakly) acute triangulation. Our bounds are nearly optimal. In addition, for d = 1 and 2 and we analyze a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal and error bounds for d = 1. For this case we also present some numerical results. Received July 4, 1996 / Revised version received December 18, 1997     

Convergence of Finite Element Approximations and Multilevel Linearization for Ginzburg–Landau Model of d-Wave Superconductors

In this paper, we consider the finite element approximations of a recently proposed Ginzburg–Landau-type model for d-wave superconductors. In contrast to the conventional Ginzburg–Landau model the scalar complex valued order-parameter is replaced by a multicomponent complex order-parameter and the free energy is modified according to the d-wave paring symmetry. Convergence and optimal error estimates and some superconvergent estimates for the derivatives are derived. Furthermore, we propose a multilevel linearization procedure to solve the nonlinear systems. It is proved that the optimal error estimates and superconvergence for the derivatives are preserved by the multilevel linearization algorithm.     

Elliptic control problems with gradient constraints—variational discrete versus piecewise constant controls

We consider an elliptic optimal control problem with pointwise bounds on the gradient of the state. To guarantee the required regularity of the state we include the L ^r -norm of the control in our cost functional with r>d (d=2,3). We investigate variational discretization of the control problem (Hinze in Comput. Optim. Appl. 30:45–63, 2005) as well as piecewise constant approximations of the control. In both cases we use standard piecewise linear and continuous finite elements for the discretization of the state. Pointwise bounds on the gradient of the discrete state are enforced element-wise. Error bounds for control and state are obtained in two and three space dimensions depending on the value of r.     

Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite difference equations with random, possibly non symmetric coe?cients. Under the assumption that the coe?cients are stationary and ergodic in the quantitative form of a logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d≥2. Similar estimates have recently been obtained in the special case of diagonal coe?cients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green’s function in weighted spaces. In the critical case d = 2, our argument for the estimate on the gradient of the elliptic Green’s function uses a Calderón–Zygmund estimate in discrete weighted spaces, which we state and prove. As applications, we provide a quantitative two-scale expansion and a quantitative approximation of the homogenized coe?cients.     

High dimensional polynomial interpolation on sparse grids

We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without certain time-step restrictions. The key is to estimate directly the solution bounds in the H²-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence and optimal error estimates, respectively, of the proposed fully discrete schemes.

     

A posteriori error estimates of constrained optimal control problem governed by convection diffusion equations

In this paper, we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations. The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L ²-norm and for state y and costate p measured in energy norm. Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.      

On the Convergence of Finite Element Method for Second Order Elliptic Interface Problems

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L ² and H ¹ norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established.     

Sharp Integral Inequalities of the Hermite–Hadamard Type

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanovi and Pecari , and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning L^p estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds.     

Cubature formulas for function spaces with moderate smoothness

We construct simple algorithms for high-dimensional numerical integration of function classes with moderate smoothness. These classes consist of square-integrable functions over the d-dimensional unit cube whose coefficients with respect to certain multiwavelet expansions decay rapidly. Such a class contains discontinuous functions on the one hand and, for the right choice of parameters, the quite natural d-fold tensor product of a Sobolev space H^s[0,1] on the other hand.The algorithms are based on one-dimensional quadrature rules appropriate for the integration of the particular wavelets under consideration and on Smolyak's construction. We provide upper bounds for the worst-case error of our cubature rule in terms of the number of function calls. We additionally prove lower bounds showing that our method is optimal in dimension d=1 and almost optimal (up to logarithmic factors) in higher dimensions. We perform numerical tests which allow the comparison with other cubature methods.     

A posteriori error estimates for a coupled wave system with a local damping

We study a finite element method applied to a system of coupled wave equations in a bounded smooth domain in \mathbbR^d {\mathbb{R}^d} , d = 1, 2, 3, associated with a locally distributed damping function. We start with a spatially continuous finite element formulation allowing jump discontinuities in time. This approach yields, L ₂(L ₂) and L _∞(L ₂), a posteriori error estimates in terms of weighted residuals of the system. The proof of the a posteriori error estimates is based on the strong stability estimates for the corresponding adjoint equations. Optimal convergence rates are derived upon the maximal available regularity of the exact solution and justified through numerical examples. Bibliography: 14 titles. Illustrations: 4 figures.     

Estimates for the Small Ball Probabilities of the Fractional Brownian Sheet

We obtain some new estimates for the small ball behavior of the d-dimensional fractional Brownian sheet under Hölder and Orlicz norms. For d=2, these bounds are sharp for the Orlicz and the sup-norm. In addition, we give bounds for the Kolmogorov and entropy numbers of some operators satisfying an L ₂-Hölder-type condition.     

An experimental survey of a posteriori Courant finite element error control for the Poisson equation

This comparison of some a posteriori error estimators aims at empirical evidence for a ranking of their performance for a Poisson model problem with conforming lowest order finite element discretizations. Modified residual-based error estimates compete with averaging techniques and two estimators based on local problem solving. Multiplicative constants are involved to achieve guaranteed upper and lower energy error bounds up to higher order terms. The optimal strategy combines various estimators.     

Finite element methods for semilinear elliptic problems with smooth interfaces

The purpose of this paper is to study the finite element method for second order semilinear elliptic interface problems in two dimensional convex polygonal domains. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with straight interface triangles [Numer. Math., 79 (1998), pp. 175–202]. For a finite element discretization based on a mesh which involve the approximation of the interface, optimal order error estimates in L ² and H ¹-norms are proved for linear elliptic interface problem under practical regularity assumptions of the true solution. Then an extension to the semilinear problem is also considered and optimal error estimate in H ¹ norm is achieved.     

Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

Summary.  We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u _t −Δu+ɛ⁻² f(u)=0 arising from phase transition in materials science, where ɛ is a small parameter known as an ``interaction length'. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ɛ. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u ₀ . In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow. Received April 30, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10 Correspondence to: A. Prohl     

A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state

We analyze a finite element approximation of an elliptic optimal control problem with pointwise bounds on the gradient of the state variable. We derive convergence rates if the control space is discretized implicitly by the state equation. In contrast to prior work we obtain these results directly from classical results for the W ^1,∞-error of the finite element projection, without using adjoint information. If the control space is discretized directly, we first prove a regularity result for the optimal control to control the approximation error, based on which we then obtain analogous convergence rates.     

An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations

We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L ²- and H ¹-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition.     

Stable enrichment and local preconditioning in the particle-partition of unity method

This paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods. In particular we focus on the particle-partition of unity method (PPUM) yet the presented results hold for any partition of unity based enrichment scheme. The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well-conditioned basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points. We obtain optimal error bounds for an enriched PPUM discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super-convergence. The results of our numerical experiments clearly show that our enriched PPUM with local preconditioning recovers the optimal convergence rate of O(h ^p ) of the uniform h-version globally. For the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of O(h ^p+δ ) with d 3 \frac12{\delta\geq\frac{1}{2}} for p = 1. The convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation.