A least-squares virtual element method for second-order elliptic problems

Abstract In this paper, a least-squares virtual element method is presented for approximating the vector and scalar variables of second-order elliptic problems. The H ( div ) -conforming and scalar-conforming virtual elements are used to approximate the vector and scalar variables, respectively. The method allows the use of very general polygonal meshes and leads to a symmetric positive definite system. The optimal a priori error estimates are established for the vector variable in H ( div ) norm and the scalar variable in H 1 norm. A simple a posteriori error estimator is also presented and proved to be reliable and efficient. The virtual element method handles the hanging nodes naturally, thus the local mesh post-processing to remove hanging nodes is not required. Numerical experiments are conducted to verify the accuracy of the method, and show the effectiveness and flexibility of the adaptive strategy driven by the proposed estimator and suitable mesh refinement strategy.