Mixed variational formulations of virtual elements for the polyharmonic operator (-)ⁿ

In this work we will present a method of virtual elements to approximate the solution of a polyharmonic problem (-Delta)???????????? = ??????. We will consider ?????? + 1 auxiliary unknowns when ?????? = 2 ?????? + 1, and ?????? auxiliary unknowns for ?????? = 2 ??????. In the first case (?????? = 2 ?????? + 1), we will solve ?????? fourth order problems and a second order one. In the even case, only ?????? fourth-order problems have to be solved. Virtual element conforming discretizations are written for each fourth -order problem in ??????1, and a ??????0-virtual element method is established for the second -order problem. We also carry out the error analysis for both cases. Finally, we report a series of numerical tests to verify the performance of numerical scheme.