Mixed variational formulations of virtual elements for the polyharmonic operator (−Δ)

In this work we will present a method of virtual elements to approximate the solution of a polyharmonic problem (−Δ)nu=g. We will consider m+1 auxiliary unknowns when n=2m+1, and m auxiliary unknowns for n=2m. In the first case (n=2m+1), we will solve m fourth order problems and a second order one. In the even case, only m fourth-order problems have to be solved. Virtual element conforming discretizations are written for each fourth-order problem in C1, and a C0-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.