Abstract It is shown how the polynomial splines defined on the curved triangulations introduced in our recent paper Larry L. Schumaker, A. Yu. (Comput. Aided Geom. Design. 92:102050, 2022) can be used to solve elliptic PDEs defined on curved planar domains. The approach is similar to that used in Larry L. Schumaker (J. Sci. Comp. 80:1369–1394, 2019), but does not require immersing the domain of interest in a larger computational domain. The methods are easy to implement using Bernstein–Bézier representations of the splines, and the solutions are computed by solving certain penalized least-squares problems. A number of numerical examples are included to illustrate the performance of the methods.
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