Some new nonconforming virtual element methods for the elasticity and stokes problems on polygonal meshes

Abstract

Recently, numerical methods for solving partial differential equations on general polygonal and polyhedral meshes have received a huge attention due to the geometric flexibility and its appearance in various applications. The virtual element method is the one of such methods, and it is regarded as a generalization of the finite element method to the general polygonal and polyhedral meshes. In this dissertation, we develop some new nonconforming virtual element methods for the two-dimensional linear elasticity and Stokes problems. For the elasticity and Stokes problems, one of the main difficulties is to design the discrete space satisfying both the Korn’s inequality and the inf-sup condition, which is especially hard for the lowestorder case. We present two kinds of lowest-order nonconforming virtual element methods for such problems. The first one uses the lowest-order nonconforming virtual element space with interior jump penalties along the interelement boundaries, while the second one uses the conforming virtual element space for one component of the displacement vector and fluid velocity and the nonconforming virtual element space for the other. These methods can be seen as extensions of the finite element methods suggested in [74] and [79], respectively. We prove the well-posedness, locking-free property and optimal convergence of our methods. On the other hand, we also present a construction of a discrete divergence-free basis in the arbitraryorder nonconforming virtual elements for the Stokes problem. Using this basis, we can eliminate the pressure variable from the mixed system and obtain a reduced symmetric positive definite system, which can be solved by more efficient methods such as the conjugate gradient method, Cholesky factorization, and multigrid method. By some numerical experiments comparing the CPU times, we observe that our method is more efficient than the original method.