Mathematics Colloquia and Seminars

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We present a framework for general higher-order finite volume cut-cell discretizations of PDE's on arbitrary domains. Our approach uses geometric moments and boundary conditions to derive polynomial fits, from which flux stencils of the required accuracy are obtained. Away from the embedded boundary, standard finite volume stencils are used with block-structured adaptive mesh refinement, making the algorithm very efficient. We are able to analyze the accuracy and stability of the approach with respect to arbitrarily small cut cells, however guaranteeing the stability of discretizations is a challenge. We will review some of the historic approaches to this problem, as well as more recent results and future research directions.