A family of stabilizer-free virtual elements on triangular meshes

09/18/2023

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by   Xuejun Xu, et al.

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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.