Adaptive reduced basis numerical homogenization methods

Assyr Abdulle
École Polytechnique Fédérale de Lausanne (EPFL)

A posteriori error control is an essential tool in numerical homogenization. Adaptive methods not only provide a criterion that indicates whether a certain prescribed accuracy is met, but also estimate local errors that allow to drive a mesh refinement that equi-distributes the approximation error
by refining the mesh in the region where singularity, e.g., crack, in the solution or in the domain occur.
Standard numerical homogenization methods require the resolution of a large number of micro problems with increasing degrees of freedom as the mesh of the physical domain is refined. Departing from the classical approach consisting in solving micro problems on sampling domains at each quadrature points of a macro FEM with numerical integration, we show that interpolation techniques based on the reduced basis methodology (based on an offline-online strategy) allow to design an efficient numerical method relying only on a small number of accurately computed micro solutions. A priori and a posteriori error analysis will be discussed.

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