Piecewise Polynomials Of Degree Research Articles

Error Estimates and Adaptive Time-Step Control for a Class of One-Step Methods for Stiff Ordinary Differential Equations

We prove new optimal a priori error estimates for a class of implicit one-step methods for stiff ordinary differential equations obtained by using the discontinuous Galerkin method with piecewise polynomials of degree zero and one. Starting from these estimates we propose a new algorithm for automatic time-step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations.

Finite element methods are not always optimal

Consider a regularly elliptic 2mth order boundary value problem Lu = f with f ϵ H r( Ω) , where r $ ̄ m . In previous work, we showed that the finite element method (FEM) using piecewise polynomials of degree k is asymptotically optimal when k ≥ 2 m − 1 + r. In this paper, we show that the FEM is not asymptotically optimal when this inequality is violated. However, there exists an algorithm, called the spline algorithm, which uses the same information as the FEM and is optimal. Moreover, the error of the finite element method can be arbitrarily larger than the error of the spline algorithm. We also obtain a necessary and sufficient condition for a Galerkin method (or a generalized Galerkin method) to be a spline algorithm.

Asymptotically optimal selection of a piecewise polynomial estimator of a regression function

Let ( X, Y) be a pair of random variables such that X = ( X 1,…, X d ) ranges over a nondegenerate compact d-dimensional interval C and Y is real-valued. Let the conditional distribution of Y given X have mean θ( X) and satisfy an appropriate moment condition. It is assumed that the distribution of X is absolutely continuous and its density is bounded away from zero and infinity on C. Without loss of generality let C be the unit cube. Consider an estimator of θ having the form of a piecewise polynomial of degree k n based on m n d cubes of length 1 m n , where the m n d ( d k n + d ) coefficients are chosen by the method of least squares based on a random sample of size n from the distribution of ( X, Y). Let ( k n , m n ) be chosen by the FPE procedure. It is shown that the indicated estimator has an asymptotically minimal squared error of prediction if θ is not of the form of piecewise polynomial.

Error Estimates and Automatic Time Step Control for Nonlinear Parabolic Problems, I

In this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem. The method for time discretization is the so-called Discontinuous Galerkin method based on using piecewise polynomials of degree $q \geqq 0$. We consider in this note the case $q = 0$ corresponding to a variant of the backward Euler method. We prove a new almost optimal error estimate and present a related new algorithm for automatic time step control. This algorithm is very simple but yet is efficient and gives control of the global error.

Superconvergent Approximations to the Solution of a Boundary Integral Equation on Polygonal Domains

The solution to the interior Dirichlet problem for Laplace’s equation in a two-dimensional domain is the double layer potential of a double layer distribution satisfying a second kind boundary integral equation. This may be solved numerically by Galerkin’s method using piecewise polynomials of degree r. The iterated Galerkin solution may then be calculated, which gives an order $2r + 2$ approximation if the domain is smooth. However if the domain is polygonal, the corners cause singularities which degrade the order of convergence. Here it is shown that a suitable grading of the mesh near the corners almost restores the rate of convergence when the error is measured in the uniform norm. We then describe a second means of calculating a higher order approximation from the Galerkin solution. This is cheaper to calculate than the iterated Galerkin solution, but maintains the same order of convergence.

Discontinuous Polynomial Approximations in the Theory of One-Step, Hybrid and Multistep Methods for Nonlinear Ordinary Differential Equations

This paper studies the approximation of the solution of nonlinear ordinary differen- tial equations by (discontinuous) piecewise polynomials of degree K and traces at the nodes of discretization. A mesh-dependent variational framework underlying this discontinuous ap- proximation is derived. Several families of one-step, hybrid and multistep schemes are obtained. It is shown that the convergence rate in the L2-norm is K + 1. The nodal-conver- gence rate can go up to 2 K + 2, depending on the particular scheme under consideration. The mesh-dependent variational framework introduced here is of special interest in the approxi- mation of the solution of optimal control problems governed by differential equations. 1. Introduction. The object of this paper is the study of (discontinuous) piecewise polynomial approximations to the solution of systems of nonlinear ordinary dif-

Analyses of Spline Collocation Methods for Parabolic and Hyperbolic Problems in Two Space Variables

Optimal order error estimates are derived for the continuous-time collocation method and a discrete-time collocation method (the Crank Nicolson collocation method) for approximating the solution of a semilinear parabolic initial-boundary value problem in $R \times ( {0,T} ]$, where R is the unit square. In each case, at each time level, the approximation is a $C^1 $ piecewise polynomial of degree $r \geqq 3$ defined by collocation at Gauss points in R. Optimal order error estimates are also derived for a family of discrete-time collocation methods for a semilinear second order hyperbolic initial-boundary value problem in $R \times ( {0,T} ]$.

AC 0-collocation-like method for two-point boundary value problems

A method of a collocation type based onC0-piecewise polynomial spaces is presented for a two-point boundary value problem of the second order. The method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forC1-collocation methods. If the differential operator is symmetric, a modification of this method leads to a symmetric system of linear equations. It is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr, the method is stable and convergent with orderhr inH1-norm. A similar symmetric modification forC0-colloction-finite element method [7] is also obtained. Superconvergence at the nodes is established.

Discontinuous polynomial approximations in the theory of one-step, hybrid and multistep methods for nonlinear ordinary differential equations

This paper studies the approximation of the solution of nonlinear ordinary differential equations by (discontinuous) piecewise polynomials of degree K and traces at the nodes of discretization. A mesh-dependent variational framework underlying this discontinuous approximation is derived. Several families of one-step, hybrid and multistep schemes are obtained. It is shown that the convergence rate in the L 2 {L^2} -norm is K + 1 K + 1 . The nodal-convergence rate can go up to 2 K + 2 2K + 2 , depending on the particular scheme under consideration. The mesh-dependent variational framework introduced here is of special interest in the approximation of the solution of optimal control problems governed by differential equations.

High-order local rate of convergence by mesh-refinement in the finite element method

We seek approximations of the solution u of the Neumann problem for the equation L u = f Lu = f in Ω \Omega with special emphasis on high-order accuracy at a given point x 0 ∈ Ω ¯ {x_0} \in \bar \Omega . Here Ω \Omega is a bounded domain in R N ( N ⩾ 2 ) {R^N}(N \geqslant 2) with smooth boundary, and L is a second-order, uniformly elliptic, differential operator with smooth coefficients. An approximate solution u h {u_h} is determined by the standard Galerkin method in a space of continuous piecewise polynomials of degree at most r − 1 r - 1 on a partition Δ h ( x 0 , α ) {\Delta _h}({x_0},\alpha ) of Ω \Omega . Here h is a global mesh-size parameter, and α \alpha is the degree of a certain systematic refinement of the mesh around the given point x 0 {x_0} , where larger α \alpha ’s mean finer mesh, and α = 0 \alpha = 0 corresponds to the quasi-uniform case with no refinement. It is proved that, for suitable (sufficiently large) α \alpha ’s the high-order error estimate ( u − u h ) ( x 0 ) = O ( h 2 r − 2 ) (u - {u_h})({x_0}) = O({h^{2r - 2}}) holds. A corresponding estimate with the same order of convergence is obtained for the first-order derivatives of u − u h u - {u_h} . These estimates are sharp in the sense that the required degree of refinement in each case is essentially the same as is needed for the local approximation to this order near x 0 {x_0} . For the estimates to hold, it is sufficient that the exact solution u have derivatives to the rth order which are bounded close to x 0 {x_0} and square integrable in the rest of Ω \Omega . The proof of this uses high-order negative-norm estimates of u − u h u - {u_h} . The number of elements in the considered partitions is of the same order as in the corresponding quasi-uniform ones. Applications of the results to other types of boundary value problems are indicated.

A Sequence of Piecewise Orthogonal Polynomials

In this paper we study the construction of an orthogonal sequence $(U_{k,n}^{(i)} )$ of piecewise polynomials of degree k which is complete in $L_2 $. Explicit expressions of $(U_{k,n}^{(i)} )$ for $k = 1,2,3$ are given. We also study the sign-change properties of this sequence and consider the convergence of the corresponding Fourier series. The results generalize those obtained earlier for Walsh system.

On the optimal approximation rates for criss-cross finite element spaces

Let Δ denote the triangulation of the plane obtained by multi-integer translates of the four lines x=0, y=0, x= y and x=− y. By l k, h μ we mean the space of all piecewise polynomials of degree ⩽ k with respect to the scaled triangulation hΔ having continuous partial derivatives of order ⩽μ on R 2 . We show that the approximation properties of l k, h μ are completely governed by those of the space spanned by the translates of all so called box splines contained in l k,h μ . Combining this fact with Fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where μ is the largest degree of smoothness for which these spaces are dense as h tends to zero. Furthermore, we study the question of local linear dependence of the translates of the box splines for the above criss-cross triangulations.

Rubel's universal differential equation

Fourth-order differential equations such as 16y'(m)y'(2) - 32y(m)y(n)y' + 17y(0(3) ) = 0 are developed. It is shown that the equation is "universal" in the sense that any continuous function can be approximated with arbitrary accuracy over the whole x axis by a solution y(x) of the equation. This solution is a piecewise polynomial of degree 9 and of class C(4).

A $C^1 $ Finite Element Collocation Method for Elliptic Equations

Collocation at Gaussian quadrature points as a means of determining a $C^1 $ finite element approximation to the solution of a linear elliptic boundary value problem on a square is studied. Optimal order $L^2 $ and $H^1 $ error estimates are established for approximation in a function space consisting of tensorproducts of $C^1 $ piecewise polynomials of degree not greater that r, where $r \geqq 3$.

A weak discrete maximum principle and stability of the finite element method in 𝐿_{∞} on plane polygonal domains. I

Let Ω \Omega be a polygonal domain in the plane and S r h ( Ω ) S_r^h(\Omega ) denote the finite element space of continuous piecewise polynomials of degree ⩽ r − 1 ( r ⩾ 2 ) \leqslant r - 1\;(r \geqslant 2) defined on a quasi-uniform triangulation of Ω \Omega (with triangles roughly of size h). It is shown that if u h ∈ S r h ( Ω ) {u_h} \in S_r^h(\Omega ) is a "discrete harmonic function" then an a priori estimate (a weak maximum principle) of the form \[ ‖ u h ‖ L ∞ ( Ω ) ⩽ C ‖ u h ‖ L ∞ ( ∂ Ω ) {\left \| {{u_h}} \right \|_{{L_\infty }(\Omega )}} \leqslant C{\left \| {{u_h}} \right \|_{{L_\infty }(\partial \Omega )}} \] holds. Now let u be a continuous function on Ω ¯ \bar \Omega and u h {u_h} be the usual finite element projection of u into S r h ( Ω ) S_r^h(\Omega ) (with u h {u_h} interpolating u at the boundary nodes). It is shown that for any χ ∈ S r h ( Ω ) \chi \in S_r^h(\Omega ) \[ ‖ u − u h ‖ L ∞ ( Ω ) ⩽ c ( ln ⁡ 1 h ) r ¯ ‖ u − χ ‖ L ∞ ( Ω ) , where r ¯ = { 1 a m p ; if r = 2 , 0 a m p ; if r ⩾ 3. {\left \| {u - {u_h}} \right \|_{{L_\infty }(\Omega )}} \leqslant c{\left ( {\ln \frac {1}{h}} \right )^{\bar r}}{\left \| {u - \chi } \right \|_{{L_\infty }(\Omega )}},\quad {\text {where}}\;\bar r = \left \{ {\begin {array}{*{20}{c}} 1 & {{\text {if}}\;r = 2,} \\ 0 & {{\text {if}}\;r \geqslant 3.} \\ \end {array} } \right . \] This says that (modulo a logarithm for r = 2 r = 2 ) the finite element method is bounded in L ∞ {L_\infty } on plane polygonal domains.

Higher-order convergence for a finite element method for the tricomi problem

We consider the numerical solution of the first-order-system version of the Tricomi problem. A new Galerkin method, using differing spaces of test and trial functions, is presented. We show that the error in the L2 norm is O(h k ) if finite element subspaces consisting of piecewise polynomials of degree k are used. For the case k = 1 of piecewise linear elements on triangles, we show that the L2-rate of convergence may be increased from 0(h) to 0(h3/2)

Simple spline-function equations for fracture mechanics calculations

The paper presents simple spline-function equations for fracture mechanics calculations. A spline function is a sequence of piecewise polynomials of degree n greater than 1 whose coefficients are such that the function and its first n-1 derivatives are continuous. Second-degree spline equations are presented for the compact, three point bend, and crack-line wedge-loaded specimens. Some expressions can be used directly, so that for a cyclic crack propagation test using a compact specimen, the equation given allows the cracklength to be calculated from the slope of the load-displacement curve. For an R-curve test, equations allow the crack length and stress intensity factor to be calculated from the displacement and the displacement ratio.

An Approximation Procedure for Nonsymmetric, Nonlinear Hyperbolic Systems with Integral Boundary Conditions

Optimal order error estimates for two linearizations of a collocation process using tensor products of continuous, piecewise linear functions in space and time are derived for a class of nonsymmetric, nonlinear hyperbolic systems with integral boundary conditions. This procedure is a variant of the so-called box scheme. Error estimates are also derived for a generalization of these procedures to collocation based on continuous, piecewise polynomials of degree r in space tensored with continuous, piecewise linear functions in time.

Finite Element Approximation for First Order Systems

In this paper we examine a finite element procedure for the numerical approximation of the solution of certain classes of boundary value problems for first order partial differential equations. Our first result shows, under.a weak regularity assumption, that the error in the $L_2 $ norm is $O(h^{K - 1} )$ for piecewise polynomials of degree $K - 1$; thus, convergence is not optimal in $L_2 $. We show by an example that there is, in fact, a loss in $L_2 $ norm. However, under stronger regularity assumptions, which apply primarily to elliptic systems, we show that convergence in $L_2 $ is optimal, e.g., with piecewise linear elements the error is of $O(h^2 )$. Numerical examples indicating the effectiveness of the method are given.

Higher order methods for a singularly perturbed problem

A finite element method using piecewise polynomials of degree ?k is used to approximate the problem ?u?+u?=f, ?>0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h k+1) are obtained uniformly in ?,h the maximum stepsize, fork?2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.