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We present a PDE-based approach for the multidimensional extrapolation ofsmooth scalar quantities across interfaces with kinks and regions of highcurvature. Unlike the commonly used method of [2] in which normal derivativesare extrapolated, the proposed approach is based on the extrapolation andweighting of Cartesian derivatives. As a result, second- and third-orderaccurate extensions in the $L^\infty$ norm are obtained with linear andquadratic extrapolations, respectively, even in the presence of sharp geometricfeatures. The accuracy of the method is demonstrated on a number of examples intwo and three spatial dimensions and compared to the approach of [2]. Theimportance of accurate extrapolation near sharp geometric features ishighlighted on an example of solving the diffusion equation on evolvingdomains.

PDE-Based Multidimensional Extrapolation of Scalar Fields over Interfaces with Kinks and High Curvatures