Contents 1 Leibniz's vision 2 Computability and Predictability 3 The Finite-element method 4 Adaptive computational methods ...

this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is devoted to the derivation of weighted a posteriori error bounds for linear functionals of the solution. Finally, in Section 4 we present some numerical examples to demonstrate the performance of the resulting adaptive finite element algorithm. 2 Model problem and discretisation Given an open bounded polyhedral domain fl in lI n, n _> 1, with boundary 0fl, we consider the following problem: find u: fl --> lI m, m _> 1, such that div(u) = 0 in , (2.1) where, ,: m __> mxn is continuously differentiable. We assume that the system of conservation laws (2.1) may be supplemented by appropriate initial/boundary conditions. For example, assuming that B(u, y) := EiL1 biVu'(u) has m real eigenvalues and a complete set of linearly independent eigenvectors for all y = (yl,---, Yn) C n; then at inflow/outflow boundaries, we require that B-(u, n)(u- g) = 0, where n denotes the unit outward normal vector to 0fl, B-(u, n) is the negative part of B(u, n) and g is a (given) real-valued function. To formulate the discontinuous Galerkin finite element method (DGFEM, for short) for (2.1), we first introduce some notation. Let 7 = {n} be an admissible subdivision of fl into open element domains n; here h is a piecewise constant mesh function with h(x) = diam(n) 2 Houston e al. when x is in element n. For p Iq0, we define the following finite element space n, -- {v [L()]": vl [%(n)] " Vn }, where Pp(n) denotes the set of polynomials of degree at most p over n. Given that v [Hi(n)] m for each n...

We outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of Navier-Stokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid V-cycle for the scalar convection-diffusion operator, and a multigrid V-cycle for a pressure Poisson operator. We present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid, robust with respect to the time-step, and only deteriorates very slowly as the Reynolds number is increased.

We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite element method. The strong stability estimate uses the entropy condition for the system in an essential way. 1. INTRODUCTION A Basic Problem One of the basic unsolved problems of computational fluid mechanics may be formulated as follows: Construct an algorithm for the numerical solution of the (compressible or incompressible) Navier-Stokes equations such that the error between the exact and computed solution in a given norm may be guaranteed to be below a given tolerance and such that (under suitable comparison) the computational work is nearly minimal. The problem clearly has two components, reliability and efficiency; the error s...

Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients. 1.

During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical tools which are used in the development of convergence theories for these algorithms. These include classical compactness arguments (based on BV a priori estimates), the use of compensated compactness arguments (based on H^-1-compact entropy production), measure valued solutions (measured by their negative entropy production), and finally, we highlight the most recent addition to this bag of analytical tools -- the use of averaging lemmas which yield new compactness and regularity results for nonlinear conservation laws and related equations. We demonstrate how these analytical tools are used in the convergence analysis of approximate solutions for hyperbolic conservation laws and related equations. Our discussion includes examples of Total Variation Diminishing (TVD) finite-difference schemes; error estimates derived from the one-sided stability of Godunov-type methods for convex conservation laws (and their multidimensional analogue -- viscosity solutions of demi-concave Hamilton-Jacobi equations); we outline, in the one-dimensional case, the convergence proof of finite-element streamline-diffusion and spectral viscosity schemes based on the div-curl lemma; we also address the questions of convergence and error estimates for multidimensional finite-volume schemes on non-rectangular grids; and finally, we indicate the convergence of approximate solutions with underlying kinetic formulation, e.g., finite-volume and relaxation schemes, once their regularizing effect is quantified in terms of the averaging lemma.

This work is a natural extension of the work done in Part II of this series. A new partition of unity --- the synchronized reproducing kernel (SRK) interpolant---is proposed within the framework of moving least square reproducing kernel representation. It is a further development and generalization of the reproducing kernel particle method (RKPM), which demonstrates some superior computational capability in multiple scale numerical simulations. To form such an interpolant, a class of new wavelet functions are introduced in an unconventional way, and they form an independent sequence that is referred to as the wavelet packet. By choosing different combinations in the wavelet series expansion, the desirable synchronized convergence effect in interpolation can be achieved. Based upon the built-in consistency conditions, the differential consistency conditions for the wavelet functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, w...

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation c t +r (uf(c)) r (Drc)+c = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L¹-norm, independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical

Contents 1 Introduction and Notations 1 2 Moving Boundary Problems 9 2.1 Flow in a Channel with a Moving Indentation : : : : : : : : : 9 2.2 Flow in a Water Pump : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Time Discretization : : : : : : : : : : : : : : : : : : : : : : : : 21 2.3.1 Investigation of the continuous problem : : : : : : : : : 21 2.3.2 Semi--discretization : : : : : : : : : : : : : : : : : : : : 25 2.3.3 Full discretization : : : : : : : : : : : : : : : : : : : : : 29 3 Adaptive Finite Elements 31 3.1 Adaptive Algorithm : : : : : : : : : : : : : : : : : : : : : : : : 32 3.2 A residual-based error estimator : : : : : : : : : : : : : : : : : 37 3.3 Multigrid method on locally refined meshes : : : : : : : : : : : 40 4 Error estimators for the Stokes Equations 49 4.1 Discretization of the

In this paper we derive an a posteriori error bound for the Lagrange–Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Based on this a posteriori bound, we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a user-defined tolerance.