Toward error estimates for general space-time discretizations of the advection equation

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

A Comparison Between Differential Equation Solver Suites In MATLAB, R, Julia, Python, C, Mathematica, Maple, and Fortran

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.

A generalized adaptive finite element analysis of laminated plates

In this paper we study the error estimates to sufficiently smooth solutions of symmetrizable systems of conservation laws for the Runge–Kutta discontinuous Galerkin (RKDG) method. Time discretization is the second‐order explicit TVD (total variation diminishing) Runge–Kutta method, and the $\mathbb{P}^k$ (piecewise polynomial) finite element is used. When $k=1$ (piecewise linear finite element), the error estimate is obtained under the usual CFL condition $\dt\leq \beta h$ for nonlinear systems in one dimension and for linear systems in multiple space dimensions. Here, h is the maximum element length, τ is the time step, and β is a positive constant independent of h and τ. Error estimates for $\mathbb{P}^k$ finite elements with $k>1$ are obtained under a more restrictive CFL condition.

This article discusses numerical errors in unsteady flow simulations, which may include round-off, statistical, iterative, and time and space discretization errors. The estimation of iterative and discretization errors and the influence of the initial condition on unsteady flows that become periodic are discussed. In this latter case, the goal is to determine the simulation time required to reduce the influence of the initial condition to negligible levels. Two one-dimensional, unsteady manufactured solutions are used to illustrate the interference between the different types of numerical errors. One solution is periodic and the other includes a transient region before it reaches a steady-state. The results show that for a selected grid and time-step, statistical convergence of the periodic solution may be achieved at significant lower error levels than those of iterative and discretization errors. However, statistical convergence deteriorates when iterative convergence criteria become less demanding, grids are refined, and Courant number increased.For statistically converged solutions of the periodic flow and for the transient solution, iterative convergence criteria required to obtain a negligible influence of the iterative error when compared to the discretization error are more strict than typical values found in the open literature. More demanding criteria are required when the grid is refined and/or the Courant number is increased. When the numerical error is dominated by the iterative error, it is pointless to refine the grid and/or reduce the time-step. For solutions with a numerical error dominated by the discretization error, three different techniques are applied to illustrate how the discretization uncertainty can be estimated, using grid/time refinement studies: three data points at a fixed Courant number; five data points involving three time steps for the same grid and three grids for the same time-step; five data points including at least two grids and two time steps. The latter two techniques distinguish between space and time convergence, whereas the first one combines the effect of the two discretization errors.

Relinearization of the error transport equations for arbitrarily high-order error estimates

Users of locally-adaptive software for initial value ordinary differential equations are likely to be concerned with global errors. At the cost of extra computation, global error estimation is possible. Zadunaisky's method and ‘solving for the error estimate’ are two techniques that have been successfully incorporated into Runge-Kutta algorithms. The standard error analysis for these techniques, however, does not take account of the stepsize selection mechanism. In this paper, some new results are presented which, under suitable assumptions show that these techniques are asymptotically valid when used with an adaptive, variable stepsize algorithm—the global error estimate reproduces the leading term of the global error in the limit as the error tolerance tends to zero. The analysis is also applied to Richardson extrapolation (step halving). Numerical results are provided for the technique of solving for the error estimate with several Runge-Kutta methods of Dormand, Lockyer, McGorrigan and Prince.

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

This chapter treats the history of mathematical foundation of primal FEM, especially a posteriori error estimates and adaptivity, based on functional analysis in Sobolev spaces. This is of equal importance as the creation of multifarious computational methods and techniques in engineering and computer sciences. BVPs for linear elliptic PDEs, mainly the Lamè equations for linear static elasticity are treated.Bounded residual explicit and various implicit error estimators of primal FEM were mainly developed by Babuška and Rheinboldt (1978), Bank and Weiser (1985), Babuška and Miller (1987) and Aubin (1967) and Nietsche (1977).Mechanically motivated explicit and implicit error estimators were created by Zienkiewicz and Zhu (1987), using gradient smoothing of the C 0- continuous displacements and stress recovery for which convergence and upper bound property were proven by Carstensen and Funken (2001).A variant of implicit a posteriori error estimators is the error of consitutive equations by Ladevèze et al. (1998). Equilibrated test stresses on element and patch levels are required, Ladevèze, Pelle (2005). Gradient-free formulations, e.g. by Cottereau, Díez and Huerta (2009), are also competitive. Generalizations of a priori and a posteriori error estimates, using the three-functional theorem by Prager and Synge (1947), are very useful.Goal-oriented error estimators for quantities of interest (as linear or nonlinear functionals, defined of closed finite supports) are of practical importance, Eriksson et al. (1995), Rannacher and Suttmeier (1997), Cirac and Ramm (1998), Ohnimus et al. (2001), Stein and Rüter (2004) and others. Textbooks by Verfürth (1996, 1999, 2013), Ainsworth and Oden (2000), Babuška and Strouboulis (2001), are available. Verification with prescribed error tolerances is realized with the above cited bounded error estimators and related discretization adaptivity, provided that the solution exists in the used test space.Moreover, model validation requires model adaptivity of the adequate physical and mathematical modeling which additionally needs experimental verification, requiring a posteriori model error estimators combined with discretization error estimators. Model reductions, e.g. for reinforced laminates, were treated by Oden (2002), and model expansions, e.g. for 3D boundary layers of 2D plate and shell theories by Stein and Ohnimus (1997), and Stein, Rüter and Ohnimus (2011).KeywordsFinite Element MethodError EstimatorPosteriori ErrorElement InterfacePosteriori Error EstimationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

With this chapter, the numerical part of the book begins. Here, numerical methods for initial value problems of systems of first-order differential equations are studied. Starting with the concept of discretizing differential equations, the class of Runge-Kutta methods is introduced. The Butcher schemes of a variety of Runge-Kutta methods are given. Further topics are consistency, convergence, estimation of the local discretization error, step-size control, A-stability, and stiffness.

In this paper, we consider an optimal control problem which is governed by a linear parabolic equation and is subject to state constraints pointwise in time. Optimal order error estimates are developed for a space-time finite element discretization of this problem. Numerical examples confirm the theoretical results. As a by-product of our analysis, we derive a new regularity result for the optimal control.

Adaptive Finite Element Methods for Optimal Control of Elastic Waves

In this article we summarize recent results on a priori error estimates for space-time finite element discretizations of linear-quadratic parabolic optimal control problems. We consider the following three cases: problems without inequality constraints, problems with pointwise control constraints, and problems with state constraints pointwise in time. For all cases, error estimates with respect to the temporal and to the spatial discretization parameters are derived. The results are illustrated by numerical examples.KeywordsOptimal controlparabolic equationserror estimatesfinite elementscontrol constraintsstate constraintsdiscretization error.

Error transport equation boundary conditions for the Euler and Navier–Stokes equations