semi-algebraic mode analysis for multigrid method on regular rectangular and triangular grids

In this work, a semi-algebraic mode analysis (sama) technique for multigrid waveform relaxation method applied to the finite element discretization on rectangular and regular triangular grids in two dimensions and cubic and triangular prism elements in three dimensions for the heat equation is proposed.  for all the studied cases especially for the general triangular prism element, both the stiffness and mass stencils are introduced comprehensively. moreover, several numerical examples are included to illustrate the efficiency of the convergence estimates. studying this analysis for the finite element method is more involved and more general than that finite-difference discretization since the mass matrix must be considered. the proposed analysis results are a very useful tool to study the behavior of the multigrid waveform relaxation method depending on the parameters of the problem.