Virtual Element Methods for hyperbolic problems on polygonal meshes
This work extends the Virtual Element Method to hyperbolic problems on polygonal meshes, providing theoretical convergence and stability analysis, which is incremental for numerical analysis of PDEs.
The paper develops a Virtual Element Method for hyperbolic problems (linear wave equations) on polygonal meshes, deriving convergence estimates in H^1 semi-norm and L^2 norm, and analyzing stability of fully discrete schemes using Newmark and Bathe methods. Numerical tests demonstrate practical behavior.
In the present paper we develop the Virtual Element Method for hyperbolic problems on polygonal meshes, considering the linear wave equations as our model problem. After presenting the semi-discrete scheme, we derive the convergence estimates in H^1 semi-norm and L^2 norm. Moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the Newmark method and the Bathe method. Finally we show the practical behaviour of the proposed method through a large array of numerical tests.