A multigrid algorithm for the $p$-version of the Virtual Element Method
This work provides a provably convergent multigrid solver for high-order virtual element discretizations, addressing a computational bottleneck in numerical PDEs.
The authors present a multigrid algorithm for solving linear systems from the p-version of the Virtual Element Method for 2D Poisson problems, proving uniform convergence and demonstrating its effectiveness as a preconditioner for Krylov methods.
We present a multigrid algorithm for the solution of the linear systems of equations stemming from the $p-$version of the Virtual Element discretization of a two-dimensional Poisson problem. The sequence of coarse spaces are constructed decreasing progressively the polynomial approximation degree of the Virtual Element space, as in standard $p$-multigrid schemes. The construction of the interspace operators relies on auxiliary Virtual Element spaces, where it is possible to compute higher order polynomial projectors. We prove that the multigrid scheme is uniformly convergent, provided the number of smoothing steps is chosen sufficiently large. We also demonstrate that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom that can be employed to accelerate the convergence of classical Krylov-based iterative schemes. Numerical experiments validate the theoretical results.