A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach

In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal error estimates for the Engquist-Osher scheme without using the fact (i) that the solution is uniformly bounded, (ii) that the scheme is total variation diminishing, and (iii) that the discrete semigroup associated with the scheme has the -contraction property, which guarantees an upper bound for the modulus of continuity in time of the approximate solution.

     

A posteriori error estimation for variational problems with uniformly convex functionals

The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form

where is a convex lower semicontinuous functional, is a uniformly convex functional, and are reflexive Banach spaces, and is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and, thus, can be justified via the duality theory.

     

Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in . In a sense to be discussed below these sharpen known quasi-optimal and estimates for the error on irregular quasi-uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution . We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non-smooth problems.

     

A posteriori error estimators for the first-order least-squares finite element method

In this paper, we propose a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method. In particular, we propose an a posteriori error estimator for when one is interested in where . Our a posteriori error estimators are obtained by assigning proper weight (in terms of local mesh size h_T) to the terms of the least-squares functional. An a posteriori error analysis yields reliable and efficient estimates based on residuals. Numerical examples are presented to show the effectivity of our error estimators.     

A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method

The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrödinger equation is analyzed in this paper. We show the existence of the resulting approximations and prove optimal order error estimates in These estimates are valid under weak restrictions on the space-time mesh.

     

A posteriori error estimates for Maxwell equations

Maxwell equations are posed as variational boundary value problems in the function space and are discretized by Nédélec finite elements. In Beck et al., 2000, a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasi-interpolation operators recently introduced in J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.

     

Error estimates for 3- narrow finite elements

We obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three-dimensional narrow elements. First, for the Lagrange interpolation we prove optimal error estimates, both in order and regularity, in for . For it is known that this result is not true. Applying extrapolation results we obtain an optimal order error estimate for functions sligthly more regular than . These results are valid both for tetrahedral and rectangular elements. Second, for the case of rectangular elements, we obtain optimal, in order and regularity, error estimates for an average interpolation valid for functions in with and .

     

Boundary element monotone iteration scheme for semilinear elliptic partial differential equations

The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the , , Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, ``higher than optimal order' error estimates can be obtained with respect to the mesh parameter . Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.

     

Error estimates on anisotropic elements for functions in weighted Sobolev spaces

In this paper we prove error estimates for a piecewise average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions.

Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses.

Moreover, we generalize the error estimates allowing on the right-hand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems.

Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements.

As an application we consider the approximation of a singularly perturbed reaction-diffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained.

     

A priori error estimates for numerical methods for scalar conservation laws. part ii: flux-splitting monotone schemes on irregular Cartesian grids

This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of in for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called consistent schemes, and prove that they converge to the entropy solution with the rate of in ; again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of in . We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

     

Optimal convergence for the finite element method in Campanato spaces

We prove a priori estimates and optimal error estimates for linear finite element approximations of elliptic systems in divergence form with continuous coefficients in Campanato spaces. The proofs rely on discrete analogues of the Campanato inequalities for the solution of the system, which locally measure the decay of the energy. As an application of our results we derive -estimates and give a new proof of the well-known -results of Rannacher and Scott.

     

Error estimates for scattered data interpolation on spheres

We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the -sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

     

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.

A posteriori error estimates are given in the energy norm and the -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

     

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation(NLDE) in the nonrelativistic limit regime, involving a small dimensionless parameter 0 ε≤ 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O(ε~2) and O(1) in time and space,respectively. We begin with the conservative Crank-Nicolson finite difference(CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ as well as the small parameter 0 ε≤ 1. Based on the error bound, in order to obtain ‘correct' numerical solutions in the nonrelativistic limit regime, i.e., 0 ε■ 1, the CNFD method requests the ε-scalability: τ = O(ε~3) and h= O(ε~(1/2)). Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and timesplitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O(ε~2) and h = O(1) when 0 ε■1. Extensive numerical results are reported to confirm our error estimates.     

A MUSCL method satisfying all the numerical entropy inequalities

We consider here second-order finite volume methods for one-dimensional scalar conservation laws. We give a method to determine a slope reconstruction satisfying all the exact numerical entropy inequalities. It avoids inhomogeneous slope limitations and, at least, gives a convergence rate of . It is obtained by a theory of second-order entropic projections involving values at the nodes of the grid and a variant of error estimates, which also gives new results for the first-order Engquist-Osher scheme.

     

Mixed Methods for Optimal Control Problems

In this paper, we investigate a posteriori error estimates of amixed finite elementmethod for elliptic optimal control problems with an integral constraint. The gradient for ourmethod belongs to the square integrable space instead of the classical H(div; Ω) space. The state and co-state are approximated by the P ₀ ² -P₁ (velocity–pressure) pair and the control variable is approximated by piecewise constant functions. Using duality argument method and energy method, we derive the residual a posteriori error estimates for all variables.     

Sharply localized pointwise and estimates for finite element methods for quasilinear problems

We establish pointwise and estimates for finite element methods for a class of second-order quasilinear elliptic problems defined on domains in . These estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order. Our pointwise estimates are similar to and rely on results and analysis techniques of Schatz for linear problems. We also extend estimates of Schatz and Wahlbin for pointwise differences in pointwise errors to quasilinear problems. Finally, we establish estimates for the error in , where is a subdomain. These negative norm estimates are novel for linear as well as for nonlinear problems. Our analysis heavily exploits the fact that Galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships, thus allowing easy application of properly formulated results for linear problems.

     

Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter , known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size and the time step size . In particular, it is shown that all error bounds depend on only in some lower polynomial order for small . The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

     

Pointwise error estimates of the local discontinuous Galerkin method for a second order elliptic problem

In this paper we derive some pointwise error estimates for the local discontinuous Galerkin (LDG) method for solving second-order elliptic problems in (). Our results show that the pointwise errors of both the vector and scalar approximations of the LDG method are of the same order as those obtained in the norm except for a logarithmic factor when the piecewise linear functions are used in the finite element spaces. Moreover, due to the weighted norms in the bounds, these pointwise error estimates indicate that when at least piecewise quadratic polynomials are used in the finite element spaces, the errors at any point depend very weakly on the true solution and its derivatives in the regions far away from . These localized error estimates are similar to those obtained for the standard conforming finite element method.

     

Besov regularity and new error estimates for finite volume approximations of the p-laplacian

Summary. In [1], we have constructed a family of finite volume schemes on rectangular meshes for the p-laplacian and we proved error estimates in case the exact solution lies in W^2,p. Actually, W^2,p is not a natural space for solutions of the p-laplacian in the case p>2. Indeed, for general L^p data it can be shown that the solution only belongs to the Besov space In this paper, we prove Besov kind a priori estimates on the approximate solution for any data in L^p. We then obtain new error estimates for such solutions in the case of uniform meshes