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A sparse spectral method for homogenization multiscale problems. (English) Zbl 1152.65099
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.

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
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