A family of stabilizer-free $P_k$ virtual elements are constructed on triangular meshes. When choosing an accurate and proper interpolation, the stabilizer of the virtual elements can be dropped while the quasi-optimality is kept. The interpolating space here is the space of continuous $P_k$ polynomials on the Hsieh-Clough-Tocher macro-triangle, where the macro-triangle is defined by connecting three vertices of a triangle with its barycenter. We show that such an interpolation preserves $P_k$ polynomials locally and enforces the coerciveness of the resulting bilinear form. Consequently the stabilizer-free virtual element solutions converge at the optimal order. Numerical tests are provided to confirm the theory and to be compared with existing virtual elements.

翻译：本文在三角形网格上构造了一类无稳定化的 $P_k$ 虚拟元。通过选取准确且适当的插值方式，可以在保持拟最优性的同时摒弃虚拟元的稳定子。插值空间定义为Hsieh-Clough-Tocher宏三角形上的连续 $P_k$ 多项式空间，其中宏三角形由将三角形的三个顶点与其重心相连而构成。我们证明，该插值方式在局部保持 $P_k$ 多项式性质，并确保所得双线性形式的强制性，从而使无稳定化虚拟元解以最优阶收敛。最后通过数值实验验证理论分析，并与现有虚拟元方法进行比较。