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A weighted Nitsche's method for interface problems with higher-order simplex elements
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DOI：10.1007/s00466-021-02132-z.png)
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Abstract
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We develop a numerical strategy based on a weighted Nitsche's approach to model a general class of interface problems with higher-order simplex elements. We focus attention on problems in which the jump in the field quantities across an interface is given. The presented method generalizes the weighted Nitsche's approach of Annavarapu et al. (Comput. Meth. Appl. Mech. Eng. 225-228:44-54, 2012) to higher-order simplices. Specifically, for higher-order simplex elements, we derive closed-form analytical expressions for the stabilization parameter arising in Nitsche's variational form. We also prescribe corresponding weights for the discrete fluxes in the consistency terms present in Nitsche's variational form. The prescribed choice of weights is shown to be optimal such that it minimizes the stabilization parameter while ensuring coercivity of the bilinear form. In the presence of large contrasts in material properties and mesh sizes, the proposed weighting yields better conditioned systems than the traditional Nitsche formulation by bounding the maximum eigenvalue of the discrete system from above. Further, the geometrical representation of curved interfaces is improved through a hierarchical local renement approach. Several numerical examples are presented with quadratic triangles to demonstrate the efficacy of the presented method.
Keywords:
Nitsche's method
X-FEM
Embedded interfaces
Unfitted interfaces
Interior penalty
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