A Tensor Multigrid Method for Solving Sylvester Tensor Equations

In this paper, we propose a tensor multigrid method and an iterative tensor multigrid method to solve the Sylvester tensor equations which may arise from the discretization of high order partial differential equations (PDEs). The convergence and complexity of the tensor multigrid algorithm are given. Finally, the feasibility and effectiveness of the new methods are verified by some numerical examples. Note to Practitioners—Sylvester tensor equation plays a very important role in the field of stability and control theory. It is significant to find an effective method to solve high order Sylvester tensor equations because of the large computational work. Based on the advantages of the multigrid method, this paper proposes a tensor multigrid method and an iterative tensor multigrid method, the latter aiming to maintain accuracy while potentially reducing time consumption. The experimental results show the superiority of our methods compared with the existing methods.