Localized orthogonal decomposition method for the wave equation with a continuum of scales

• This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an L2-projection. We derive explicit convergence rates of the method in the L∞(L2)-, W1,∞(L2)-and L∞(H1)-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Finite element

LOD

Multiscale method

Numerical homogenization

Wave equation

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