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：在三角形网格上构造了一族无稳定子$P_k$虚拟元。通过选取精确且合理的插值方式，可在保持拟最优性的前提下舍弃虚拟元的稳定子。本文采用的插值空间为Hsieh-Clough-Tocher宏三角形上的连续$P_k$多项式空间，其中宏三角形由连接三角形三个顶点与其重心构成。研究表明，该插值格式能局部保持$P_k$多项式空间，并确保所构造双线性形式的强制性。由此得到的无稳定子虚拟元解具有最优收敛阶。数值实验验证了理论分析结果，并与现有虚拟元方法进行了对比。