Finite element heterogeneous multiscale method for the wave equation

A finite element heterogeneous multiscale method is proposed for the wave equation with highly oscillatory coefficients. It is based on a finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective coefficients are computed on sampling domains at the micro scale within each macro finite element. Hence the computational work involved is independent of the highly heterogeneous nature of the medium at the smallest scale. Optimal error estimates in the energy norm and the L-2 norm and convergence to the homogenized solution are proved, when both the macro and the micro scales are refined simultaneously. Numerical experiments corroborate the theoretical convergence rates and illustrate the behavior of the numerical method for periodic and heterogeneous media.

Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems

Assyr Abdulle, Yun Bai

A new finite element method for the efficient discretization of elliptic homogenization problems is proposed. These problems, characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed. Recently, the finite element heterogeneous multiscale method (FE-HMM) has been proposed for such problems, based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh. Departing from the approach used in the FE-HMM, we show that interpolation techniques based on the reduced basis methodology (an offline-online strategy) allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions. This new method, called the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is significantly more efficient than the FE-HMM for high order macroscopic discretizations and for three-dimensional problems, when the repeated computation of micro problems over the whole computational domain is expensive. A priori error estimates of the RB-FE-HMM are derived. Numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method. (C) 2012 Elsevier Inc. All rights reserved.

Elsevier2012

A priori error analysis of the finite element heterogeneous multiscale method for the wave equation in heterogeneous media over long time

Assyr Abdulle, Timothée Noé Pouchon

A fully discrete a priori analysis of the finite element heterogeneous multiscale method introduced in [A. Abdulle, M. Grote, and C. Stohrer, Multiscale Model. Simul., 12, 2014, pp. 1135-1166] for the wave equation with highly oscillatory coefficients over long time is presented. A sharp a priori convergence rate for the numerical method is derived for long time intervals. The effective model over long time is a Boussinesq-type equation that has been shown to approximate the one-dimensional multiscale wave equation with epsilon-periodic coefficients up to time O(epsilon(-2)) in [A. Lamacz, Math. Models Methods Appl. Sci., 21, 2011, pp. 1871-1899]. In this paper we also revisit this result by deriving and analyzing a family of effective Boussinesq-type equations for the approximation of the multiscale wave equation that depends on the normalization chosen for certain micro functions used to define the macroscopic models.

Siam Publications2016

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Assyr Abdulle, Gilles Vilmart

A fully discrete analysis of the finite element heterogeneous multiscale method for a class of nonlinear elliptic homogenization problems of nonmonotone type is proposed. In contrast to previous results obtained for such problems in dimension

d\leq2

for the

H^1

norm and for a semi-discrete formulation [W.E, P. Ming and P. Zhang, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156], we obtain optimal convergence results for dimension

d\leq3

and for a fully discrete method, which takes into account the microscale discretization. In addition, our results are also valid for quadrilateral finite elements, optimal a-priori error estimates are obtained for the

H^1

and

L^2

norms, improved estimates are obtained for the resonance error and the Newton method used to compute the solution is shown to converge. Numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method for various nonlinear problems.

Amer Mathematical Soc2014

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Assyr Abdulle, Gilles Vilmart

d\leq2

for the

H^1

norm and for a semi-discrete formulation [W.E, P. Ming and P. Zhang, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156], we obtain optimal convergence results for dimension

d\leq3

H^1

and

L^2

Amer Mathematical Soc2014