Discontinuous Piecewise Polynomials Research Articles

Spectral refinement using a new projection method

AbstractIn this paper we consider two spectral refinement schemes, elementary and double iteration, for the approximation of eigenelements of a compact operator using a new approximating operator. We show that the new method performs better than the Galerkin, projection and Sloan methods. We obtain precise orders of convergence for the approximation of eigenelements of an integral operator with a smooth kernel using either the orthogonal projection onto a spline space or the interpolatory projection at Gauss points onto a discontinuous piecewise polynomial space. We show that in the double iteration scheme the error for the eigenvalue iterates using the new method is of the order of , where h is the mesh of the partition and k = 0, 1, 2,… denotes the step of the iteration. This order of convergence is to be compared with the orders in the Galerkin and projection methods and in the Sloan method. The error in eigenvector iterates is shown to be of the order of in the new method, in the Galerkin and projection methods and in the Sloan method. Similar improvement is observed in the case of the elementary iteration. We show that these orders of convergence are preserved in the corresponding discrete methods obtained by replacing the integration by a numerical quadrature formula. We illustrate this improvement in the order of convergence by numerical examples.

Nonconforming Mixed Finite-Element Approximations to Time-Harmonic Eddy Current Problems

We present nonconforming mixed finite-element methods for the discretization of time-harmonic eddy current problems. These methods are based on a discontinuous Galerkin approach, where the unknowns are approximated by completely discontinuous piecewise polynomials. We consider a stabilized mixed formulation involving equal-order elements and a nonstabilized variant employing mixed-order elements.

A superconvergence result for solutions of compact operator equations

Over the last 20 years, since the publication of Sloan's paper on the improvement by the iteration technique, various approaches have been proposed for post-processing the Galerkin solution of multi-dimensional second kind Fredholm Integral equation. These methods include the iterated Galerkin method proposed by Sloan, the Kantorovich method and the iterated Kantorovich method. Recently, Lin, Zhang and Yan have proposed interpolation as an alternative to the iteration technique. For an integral operator, with a smooth kernel using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree ≤r− 1, previous authors have established an orderrconvergence for the Galerkin solution and 2rfor the iterated Galerkin solution. Equivalent results have also been established for the interpolator projection at Gauss points and some interpolation post-processing technique. In this paper, a method is introduced and shown to have convergence of order 4r. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.

On the Improvement of Efficiency and Storage Requirements of the Discontinuous Galerkin Method for Aeroacoustics

AbstractFor Computational Aeroacoustics (CAA), good wave propagation properties are a crucial point for numerical schemes which solve the linearized Euler equations (LEE) in the time domain. A recently developed high order scheme with excellent wave propagation properties is the Discontinuous Galerkin (DG) method. The DG schemes are based on the Finite Element Method (FEM), using discontinuous piecewise polynomial basis functions to represent the solution. The efficiency can be highly improved for CAA applications if a quadrature‐free formulation of the DG method is used, but nevertheless the DG schemes consume much computer storage due to the Runge‐Kutta time integration procedure and the high number of degrees of freedom to be stored for each element. We managed to develop a high order single‐step quadrature‐free DG method which does not use a Runge‐Kutta time integration procedure but which applies the ADER approach of Toro, Schwartzkopff et al. and where the accuracy of the scheme can further be improved using additionally a reconstruction technique working on the degrees of freedom of the element and its neighbors.

Nonlinear piecewise polynomial approximation beyond Besov spaces

We study nonlinear n-term approximation in L p( R 2) (0< p<∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R 2 which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness spaces generated by multilevel nested triangulations. We call them B-spaces because they can be viewed as generalizations of Besov spaces. We use the B-spaces to prove Jackson and Bernstein estimates for n-term piecewise polynomial approximation and consequently characterize the corresponding approximation spaces by interpolation. We also develop methods for n-term piecewise polynomial approximation which capture the rates of the best approximation.

A New Superconvergent Projection Method for Approximate Solutions of Eigenvalue Problems

We propose here a new method based on projections for approximate solution of eigenvalue problems associated with a compact linear operator. For an integral operator with a smooth kernel using the orthogonal projection onto the space of discontinuous piecewise polynomials of degree ≤r−1, we show that the new method exhibits the error of the order of 4r for eigenvalue approximation and of the order of 3r for spectral subspace approximation. This improves upon the order 2r for eigenvalue approximation in Galerkin/Iterated Galerkin method and the orders r and 2r for spectral subspace approximation in Galerkin and the Iterated Galerkin method, respectively. We illustrate this improvement in the order of convergence by a numerical example.

A stabilized discontinuous finite element method for elliptic problems

AbstractA new finite element method is proposed and analysed for second order elliptic equations using discontinuous piecewise polynomials on a finite element partition consisting of general polygons. The new method is based on a stabilization of the well‐known primal hybrid formulation by using some least‐squares forms imposed on the boundary of each element. Two finite element schemes are presented. The first one is a non‐symmetric formulation and is absolutely stable in the sense that no parameter selection is necessary for the scheme to converge. The second one is a symmetric formulation, but is conditionally stable in that a parameter has to be selected in order to have an optimal order of convergence.Optimal‐order error estimates in some H1‐equivalence norms are established for the proposed discontinuous finite element methods. For the symmetric formulation, an optimal‐order error estimate is also derived in the L2 norm. The new method features a finite element partition consisting of general polygons as opposed to triangles or quadrilaterals in the standard finite element Galerkin method. Copyright © 2002 John Wiley &amp; Sons, Ltd.

The discontinuous Petrov–Galerkin method for elliptic problems

We propose a novel discontinuous mixed finite element formulation for the solution of second-order elliptic problems. Fully discontinuous piecewise polynomial finite element spaces are used for the trial and test functions. The discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that on the one hand enforce the weak continuity of the trial functions, and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous Galerkin methods. Static condensation is performed at the element level, leading to a solution procedure based on the sole interface unknowns. The resulting family of discontinuous dual–primal mixed (DPM) finite element methods is presented in the one and two-dimensional cases. In the one-dimensional case, we show the equivalence of the method with implicit Runge–Kutta schemes of the collocation type exhibiting optimal behavior and we develop the stability analysis. Numerical experiments in one and two dimensions demonstrate the order accuracy of the new method, confirming the results of the analysis.

Galerkin Methods in Age and Space for a Population Model with Nonlinear Diffusion

We present Galerkin methods in both the age and space variables for an age-dependent population undergoing nonlinear diffusion. The methods presented are a generalization of the methods presented in {\it A Variable Time Step Method for an Age-dependent Population Model with Nonlinear Diffusion}, where the approximation space in age was taken to be the space of piecewise constant functions. In this paper, we allow the use of discontinuous piecewise polynomial subspaces of $L^2$ as the approximation space in age. As in the piecewise constant case, we move the discretization along characteristic lines. The time variable has been left continuous. The methods are shown to be superconvergent in the age variable.

Locking-free finite elements for the Reissner-Mindlin plate

Two new families of Reissner-Mindlin triangular finite elements are analyzed. One family, generalizing an element proposed by Zienkiewicz and Lefebvre, approximates (for k > 1) the transverse displacement by continuous piecewise polynomials of degree k + 1, the rotation by continuous piecewise polynomials of degree k + 1 plus bubble functions of degree k + 3, and projects the shear stress into the space of discontinuous piecewise polynomials of degree k. The second family is similar to the first, but uses degree k rather than degree k + 1 continuous piecewise polynomials to approximate the rotation. We prove that for 2 < s < k + 1, the L 2 errors in the derivatives of the transverse displacement are bounded by Ch s and the L 2 errors in the rotation and its derivatives are bounded by Ch s min(1, ht -1 ) and Ch s-1 min(1, ht -1 ), respectively, for the first family, and by Ch s and Ch s-1 , respectively, for the second family (with C independent of the mesh size h and plate thickness t). These estimates are of optimal order for the second family, and so it is locking-free. For the first family, while the estimates for the derivatives of the transverse displacement are of optimal order, there is a deterioration of order h in the approximation of the rotation and its derivatives for t small, demonstrating locking of order h -1 . Numerical experiments using the lowest order elements of each family are presented to show their performance and the sharpness of the estimates. Additional experiments show the negative effects of eliminating the projection of the shear stress.

Semi-discrete Galerkin approximation of the single layer equation by general splines

This paper deals with a semi-discrete version of the Galerkin method for the single-layer equation in a plane, in which the outer integral is approximated by a quadrature rule. A feature of the analysis is that it does not require high precision quadrature or exceptional smoothness of the data. Instead, the assumptions on the quadrature rule are that constant functions are integrated exactly, that the rule is based on sufficiently many quadrature points to resolve the approximation space, and that the Peano constant of the rule is sufficiently small. It is then shown that the semi-discrete Galerkin approximation is well posed. For the trial and test spaces we consider quite general piecewise polynomials on quasi-uniform meshes, ranging from discontinuous piecewise polynomials to smoothest splines. Since it is not assumed that the quadrature rule integrates products of basis functions exactly, one might expect degradation in the rate of convergence. To the contrary, it is shown that the semi-discrete Galerkin approximation will converge at the same rate as the corresponding Galerkin approximation in the $H^0$ and $H^{-1}$ norms.

Analysis of an algorithm for generating locally optimal meshes for 𝐿₂ approximation by discontinuous piecewise polynomials

This paper discusses the problem of constructing a locally optimal mesh for the best L 2 L_2 approximation of a given function by discontinuous piecewise polynomials. In the one-dimensional case, it is shown that, under certain assumptions on the approximated function, Baines’ algorithm [M. J. Baines, Math. Comp., 62 (1994), pp. 645-669] for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh. The rate of convergence is investigated. A two-dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self-adjusting. Numerical results in one and two dimensions are presented.

Locking-free finite element methods for shells

We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer k. The methods are based on a nonstandard mixed formulation, and the kth method employs triangular Lagrange finite elements of degree k+2 augmented by bubble functions of degree k + 3 for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree k for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone; as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the H I norm of the displacement and ro tation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order k+1 uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants.

Compact higher order characteristic-based Euler solver for unstructured grids

Two new methods are presented for solving the Euler equations using a compact higher order polynomial reconstruction technique on unstructured grids. The methods use a characteristic -based approach with a cell-centered finite volume method. For transonic Ringleb flow, computations are performed for first-order to fourth-order accuracy and are compared with the hodograph solution. Results for a 10-deg ramp case are also presented. An analysis is performed that demonstrates that the higher order method is an order of magnitude more efficient than the lower order method in modeling the flow for moderate-to-fine error tolerances. Accuracy, speed, and memory requirements are evaluated in the efficiency study. COMPACT higher order polynomial reconstruction technique is developed that allows for higher order char- acteristic-based numerical solutions to the Euler equations on unstructured grids. This reconstruction requires an extended set of Euler equations that includes as dependent variables not only mass, momentum, and energy but also their spatial derivatives. These additional dependent variables are used in a compact polynomial reconstruction process within a finite- volume framework. The development presented follows some of the ideas in the computational fluid dynamics (CFD) litera- ture using continuous and discontinuous piecewise polynomial approximations. Continuous piecewise linear polynomials have been used in several finite volume and finite element unstructured grid methods. Jameson and Mavriplis,1 Lohner et al.,2 and others have developed continuous piecewise linear elements using an added artificial dissipation and/or flux-corrected transport terms. More recently, Oden3 has extended the linear elements by the use of higher order/? elements. Oden's work represents a compact higher order approximation of the continuous ele- ment type. Discontinuous piecewise linear polynomials have similarly been used on unstructured grids with a characteristic -based approach. Thareja et al., 4 Batina,5 Frink,6 and others have developed discontinuous piecewise linear schemes with ap- proximate Riemann solvers on unstructured grids. Earth and Frederickson7 developed a technique where polynomials of degree k are reconstructed for each cell by maintaining aver- age dependent variable properties over a neighboring set of cells in a least squares formulation. These polynomials are reconstructed using a large number of neighboring cells; there- fore, the representation is not compact. This paper describes two methods that compactly reconstruct higher order polyno- mials over each cell in a discontinuous piecewise polynomial procedure. Two methods are presented for extending the Euler equa- tion set. The first method solves the spatial derivatives of the governing integral equations. The second method follows the work of Allmaras, 8 where the spatial moments of the govern- ing equations are solved. Allmaras developed an approach

Numerical Analysis of a Bidimensional Hencky Problem Approximated by a Discontinuous Finite Element Method

A finite element method for the solution of a bidimensional elastoplastic Hencky problem is formulated and analyzed. The discrete spaces consist of discontinuous piecewise polynomial functions. Convergence results for the discrete displacements are proved. Then, error estimates in ${\bf L}^2 $-norms are stated for the stress tensors, under a piecewise ${\bf H}^2 $-regularity hypothesis that is both weaker than the one commonly used and physically admissible.

Extrapolation of the Iterated–Collocation Method for Integral Equations of the Second Kind

For integral equations of the second kind with appropriate smoothness of both kernel and exact solution, it is known that the iterated–collocation method based on discontinuous piecewise polynomials of order r, and on collocation at the Gauss points, can exhibit a uniform (superconvergent) error of order $2r$. This paper shows that the order of the iterated–collocation approximation on an arbitrary mesh can be further improved by one step of Richardson extrapolation, assuming the calculation to be repeated with each subinterval halved.

A note on conditioning, stability and collocation matrices

The concept of a well-conditioned BVP, as considered in recent literature, essentially corresponds to the concept of a uniformly stable IVP. But this terminology suggests a shift from absolute to relative quantities, which makes it more difficult to understand the conditioning of numerical discretizations, especially for higher order differential operators. In this note we wish to make it clear that the notion of well-conditioning of BVPs is not always a direct analogue of the u The condition number of a discretization of a well-conditioned BVP generally grows unboundedly when higher accuracy is desired, because it contains a discretization of a differential operator. This condition number gets large particularly rapidly if a direct discretization of a higher order differential operator is attempted. For collocation, an implementation which uses a local (Runge-Kutta) solution representation produces a much better conditioning, because the repeated numerical differentiation is avoided. Instead, the highest derivative is approximated locally by a (discontinuous) piece-wise polynomial, and local integrations yield lower solution derivatives. Upon imposing the collocation equations the highest derivative approximation can be locally eliminated, resulting in a sophisticated discretization of the corresponding first order system.

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-

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.

Collocation methods using piecewise polynomials for second kind integral equations

We consider the numerical solution of second kind Fredholm integral equations in one dimension by using the collocation method and its iterated variant. The collocation solution will be sought in a space of piecewise polynomials of order r. Superconvergence results for the iterated collocation solution are known when discontinuous piecewise polynomials are used and the collocation points are taken to be the r Gaussian points shifted to each subinterval. Here we give a corresponding superconvergence result for the iterated collocation solution when continuous piecewise polynomials with no continuity requirements on the derivatives are used. The collocation points are taken to be the knots plus the r−2 Lobatto points shifted to each subinterval. Some numerical results are also given.