In the framework of virtual element discretizazions, we address the problem of imposing non homogeneous Dirichlet boundary conditions in a weak form, both on polygonal/polyhedral domains and on two/three dimensional domains with curved boundaries. We consider a Nitsche's type method [43,41], and the stabilized formulation of the Lagrange multiplier method proposed by Barbosa and Hughes in [9]. We prove that also for the virtual element method (VEM), provided the stabilization parameter is suitably chosen (large enough for Nitsche's method and small enough for the Barbosa-Hughes Lagrange multiplier method), the resulting discrete problem is well posed, and yields convergence with optimal order on polygonal/polyhedral domains. On smooth two/three dimensional domains, we combine both methods with a projection approach similar to the one of [31]. We prove that, given a polygonal/polyhedral approximation $\Omega_h$ of the domain $\Omega$, an optimal convergence rate can be achieved by using a suitable correction depending on high order derivatives of the discrete solution along outward directions (not necessarily orthogonal) at the boundary facets of $\Omega_h$. Numerical experiments validate the theory.
翻译:在虚拟元素分化的框架内,我们解决了将非同质的二分红边界条件以弱的形式强加给多边形/圆形域和具有曲线边界的两维/三维域的问题。我们考虑了尼采型方法[4341],以及Barbosa和Hughes在[9]中提议的拉格兰梯乘数法的稳定配方。我们还证明,对于虚拟元素方法(VEM),只要稳定参数选择得当(对于Nitsche的方法来说足够大,对于Barbosa-Hughes Lagrange乘数法来说足够小),由此产生的离散问题已经很好地提出,并产生多边形/圆形域的最佳顺序。在平滑的两个/三维域,我们把这两种方法与与[31]中的一种类似预测法结合起来。我们证明,鉴于域的多角/圆形近值$\Omega_h$,通过在离散的理论上对离心模型进行适当校准,可以实现最佳的趋同率。