Speaker
Description
Goal-oriented error estimation is a powerful tool for efficient and reliable numerical simulations. It allows one to control the error for specific quantities of interest and thus, enables more reliable and efficient simulations. To compute the numerical solution, we utilize the Virtual Elements Method (VEM). The VEM can handle general polygonal meshes which allows for an easy incorporation of hanging nodes that arise due to mesh refinement. In this talk, we present the first comprehensive framework for goal-oriented error estimation in VEM. We address two key challenges: First, the VEM stabilization term introduces additional error contributions that need to be estimated. To do so, the virtual shape functions need to be approximated inside the elements. Second, the standard techniques for approximating the exact adjoint solution fail in the VEM context, especially when exploiting VEM's key advantage of handling general polygonal meshes. We address these challenges by introducing efficient approximation techniques for virtual shape functions and introduce the Gauss-Point Reconstruction Method (GPRM) to approximate the exact adjoint solution. Through various numerical experiments, we validate the effectiveness of our proposed approaches and demonstrate their application to adaptive mesh refinement procedures. The results illustrate the capabilities as well as the limitations of the proposed framework.
