Abstract
We derive and analyze finite dimensional approximations for solutions to elliptic problems with coefficients or geometries which are oscillating at a small length scale ε. We introduce problem-dependent, generalized Finite Element (FE) spaces with non-polynomial micro-shape functions that reflect the correct oscillatory behaviour of the solution. Our choice of micro-scale shape functions is motivated by representation formulas for solutions on unbounded domains, which are based on scale separation and generalized Fourier inversion formulas. Under analyticity assumptions on the input data, we prove exponential convergence of the generalized FEM based on these spaces which is robust, i.e., the rate of convergence is independent of ε.1
Research supported by the Swiss National Science Foundation under Project Number BBW 21-58754.99 and under IHP Network ‘Homogenization and Multiple Scales’ “HMS2000” of the EC.
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Dedicated to Professor Ivo Babuška at the 75th anniversary
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Matache, AM., Schwab, C. (2002). Finite Dimensional Approximations for Elliptic Problems with Rapidly Oscillating Coefficients. In: Antonić, N., van Duijn, C.J., Jäger, W., Mikelić, A. (eds) Multiscale Problems in Science and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56200-6_9
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DOI: https://doi.org/10.1007/978-3-642-56200-6_9
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