The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.

Abstract. Heterogeneous multiscale methods have been introduced by E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87-132] as a methodology for the numerical computation of problems with multiple scales. Analyses of the methods for various homogenization problems have been done by several authors. These results were obtained under the assumption that the microscopic models (the cell problems in the homogenization context) are analytically given. For numerical computations, these microscopic models have to be solved numerically. Therefore, it is important to analyze the error transmitted on the macroscale by discretizing the fine scale. We give in this paper H 1 and L 2 a priori estimates of the fully discrete heterogeneous multiscale finite element method. Numerical experiments confirm that the obtained a priori estimates are sharp. 1. Introduction. The heterogeneous multiscale methods (HMM) introduced in [6] are a general framework for the numerical modeling of problems with multiple scales. For homogenization problems in a finite element (FE) context, this method discretizes the physical problem directly by a "macroscopic finite element method (FEM)" model. The fine scale of the problem is accounted for in the element stiffness matrix calculations by solving either a unit-cell problem or a problem on a patch with a fixed, i.e., scale-independent, number of unit cells. These problems will be referred to as microproblems. So far, the study of the accuracy properties in HMM has been done assuming that the fine-scale problems were analytically given [6] . We note that in [11] macro-and microerrors were first separated and quantitatively estimated, although not in the HMM context and for unbounded domains. The analysis in [11] was also restricted to the case where the diffusion tensor a ε (x) = a(x/ε) does not depend on the macrovariable, and it cannot be easily generalized.Recently, there have been many attempts to combine microscopic and macroscopic models for the solution of multiscale or even multiphysics problems (see, for example, [7] for a discussion on these topics). The influences of the error of a micromodel at a macroscale is an important question in multiscale computations but is, in general, difficult to address. To the best of our knowledge, this paper gives the first result of such an analysis in the HMM context. Within the framework introduced by E, Ming, and Zhang [5] we give several error estimates for the macroscopic solutions, when the microproblems are discretized by a numerical method. These estimates include H 1 and L 2 error estimates between the solution of the finite element heterogeneous multiscale method (FE-HMM) and

Abstract. In this paper, a new family of fourth order Chebyshev methods (also called stabilized methods) is constructed. These methods possess nearly optimal stability regions along the negative real axis and a three-term recurrence relation. The stability properties and the high order make them suitable for large stiff problems, often space discretization of parabolic PDEs. A new code ROCK4 is proposed, illustrated at several examples, and compared to existing programs.Key words. stiff ordinary differential equations, explicit Runge-Kutta methods, orthogonal polynomials, parabolic partial differential equations AMS subject classifications. 65L20, 65M20PII. S10648275003795491. Introduction. Chebyshev methods are a class of explicit Runge-Kutta methods with extended stability domains along the negative real axis. The stability properties of these methods make them suitable for stiff problems which possess a Jacobian matrix with (possibly large) eigenvalues close to the real negative axis. Since they are explicit, Chebyshev methods avoid linear algebra difficulties and can be applied to very large problems. The main applications are parabolic PDEs when discretized by finite difference. It usually gives a large system of ODEs with a symmetric and negative definite Jacobian matrix. Thus, the eigenvalues of the discretized parabolic PDEs are real negative and, furthermore, become larger while refining the space discretization.Recently, a new strategy to construct second order Chebyshev methods has been proposed by Abdulle and Medovikov [2]. It combines the advantages of the methods introduced by Lebedev [11], [12] and van der Houwen and Sommeijer [9] (see also [14] for the latest implementation of these methods). An algorithm to construct nearly optimal stability functions along the real negative axis based on orthogonal polynomials was proposed in [2]. The advantage of using orthogonal polynomials is the three-term recurrence relation which can be used to construct the numerical methods. At the same time, choosing an appropriate weight function for these polynomials leads to a nearly optimal stability domain (see [2]).For order more than 2, the only known Chebyshev methods are those of Medovikov [10]. They are constructed upon the strategy of Lebedev-type methods: the zeros of the optimal stability polynomials are computed, and the numerical methods are based on a suitable ordering of these zeros. The drawback is that the ordering, crucial for the internal stability of the methods, depends on the degree of the polynomials and needs some art. There are also no recurrence relations. The methods of order 4 proposed in this paper are based on a three-term recurrence relation and avoid the preceding problems.

Stabilized methods (also called Chebyshev methods) are explicit Runge-Kutta methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. The aim of this paper is to show that with the use of orthogonal polynomials, we can construct nearly optimal stability polynomials of second order with a three-term recurrence relation. These polynomials can be used to construct a new numerical method, which is implemented in a code called ROCK2. This new numerical method can be seen as a combination of van der Houwen-Sommeijer-type methods and Lebedev-type methods

We introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of stochastic differential equations, independently of the weak order of accuracy of the method. We then present a systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic SDEs (Brownian and Langevin dynamics) with an accuracy independent of the weak order of the underlying method. Numerical experiments confirm our theoretical findings.

Abstract. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.

Abstract. We present a finite element method for the numerical solution of diffusion problems on rough surfaces. The problem is transformed to an elliptic homogenization problem in a two dimensional parameter domain with a rapidly oscillating diffusion tensor and source term. The finite element method is based on the heterogeneous multiscale methods of E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87-132]. For periodic surface roughness of scale ε and amplitude O(ε), the method converges at a robust rate, i.e., independent of ε, to the homogenized solution.

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