Daubechies, Ingrid; Runborg, Olof; Zou, Jing A sparse spectral method for homogenization multiscale problems. (English) Zbl 1152.65099 Multiscale Model. Simul. 6, No. 3, 711-740 (2007). The authors study the multiscale problem in the form of the parabolic partial differential equation\[ \partial_{t}u-\partial_{x}a^{\varepsilon}(t,x)\partial_{x}u=0, \quad \quad u(0,x)=f(x) \]with periodic boundary condition on \([0,2\pi]\) and \( a^{\varepsilon}(t,x+2\pi)=a^{\varepsilon}(t,x)\). The interval \( [0,2\pi]\) is divided uniformly by step \(\Delta t, t_{n}=n \Delta t\). The solution of this problem is approximated by the \(N\) lowest Fourier modes in \(t_{n}\). In the standard spectral method its coefficients are calculated using the Fourier transform. The authors suggest another operator of projection. Reviewer: Ivan Secrieru (Chişinău) Cited in 7 Documents MSC: 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65T50 Numerical methods for discrete and fast Fourier transforms 35K15 Initial value problems for second-order parabolic equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure Keywords:multiscale parabolic equation; Fourier spectral method; modified Fourier method; sublinear algorithm; homogenization PDFBibTeX XMLCite \textit{I. Daubechies} et al., Multiscale Model. Simul. 6, No. 3, 711--740 (2007; Zbl 1152.65099) Full Text: DOI