The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasi-continuum method.

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...

A high-order discontinuous Galerkin finite element discretization and p-multigrid solution procedure for the compressible Navier-Stokes equations are presented. The discretization has an element-compact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the p-multigrid solver, which relies on an element-line Jacobi smoother. The element-line Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2-D scalar convection-diffusion shows that the element-line Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higher-order is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.

In these notes an introduction is given to space-time discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The space-time DG discretization is explained in detail, including the definition of the numerical fluxes and stabilization operators necessary to maintain stable and non-oscillatory solutions. In addition, a pseudo-time integration method for the solution of the algebraic equations resulting from the DG discretization and the relation between the space-time DG method and an arbitrary Lagrangian Eulerian approach are discussed. Finally, a brief overview of some applications to aerodynamics is given.

Abstract. We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continuous Galerkin, non-conforming and a new wide class of hybridizable discontinuous Galerkin methods. The main feature of the methods in this framework is that their approximate solutions can be expressed in an element-by-element fashion in terms of an approximate trace satisfying a global weak formulation. Since the associated matrix is symmetric and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled. Finally, the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying new, extremely localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom. 1.

We propose stabilized interior penalty discontinuous Galerkin methods for the indefinite time–harmonic Maxwell system. The methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field. We prove optimal error estimates for the methods in the special case of smooth coefficients and perfectly conducting boundary using a duality approach.

In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabilizations, including jump penalty, upwinding, and Hughes–Franca type residual-based stabilizations.

The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized. This is the basis for a rapid online computation in case of multiple-simulation requests. We introduce a new offline basis-generation algorithm based on our a posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach. 1

Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG1-03035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higher-order DG. In particular, Krzysztof Fidkowski and Todd Oliver are to be acknowledged for their contributions towards the development of the flow solvers and also for providing some of the grids for the test cases demonstrated. Finally, thanks must go to thesis committee members Professors Peraire and Willcox as well as thesis readers Dr. Natalia Alexandrov and Dr. Steven Allmaras for the time they put into reading the thesis and providing the valuable feedbacks. 3 46 Adjoint approach to shape sensitivity 117 6.1 Introduction...............................

Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that involves only the original unknown variable. The proposed method is closely related to the local discontinuous Galerkin (LDG) method [B. Cockburn and C.-W. Shu, SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463], but, unlike the LDG method, the sparsity pattern of the CDG method involves only nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly offset by increased compactness and flexibility.