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Cut finite element error estimates for a class of nonliner elliptic PDEs

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conference contribution
posted on 20.04.2020, 08:17 by G. Katsouleas, E. N. Karatzas, F. Travlopanos
In the contexts of fluid–structure interaction and reduced order modeling for parametrically–dependent domains, immersed and embedded methods compare favorably to standard FEMs, providing simple and efficient schemes for the numerical approximation of PDEs in both cases of static and evolving geometries. In this note, the a priori analysis of unfitted numerical schemes with cut elements is extended beyond the realm of linear problems. More precisely, we consider the discretization of semilinear elliptic boundary value problems of the form −∆u + f1(u) = f2 with polynomial nonlinearity via the cut finite element method. Boundary conditions are enforced, using a Nitsche–type approach. To ensure stability and error estimates that are independent of the position of the boundary with respect to the mesh, the formulations are augmented with additional boundary zone ghost penalty terms. These terms act on the jumps of the normal gradients at faces associated with cut elements. A–priori error estimates are derived, while numerical examples illustrate the implementation of the method and validate the theoretical findings.



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