Continuous and discontinuous Galerkin methods with finite elements in space and time for parallel computing of viscoelastic deformation

New non-symmetric variational and discretized formulations (with space-time finite elements) are proposed for viscoelastic problems based on the continuous Galerkin method (CGM) and discontinuous Galerkin method (DGM). Viscoelastic behaviour is described by the three-parameter Malvern model, which is represented by means of internal variables. It allows to use only differential equations for the constitutive equations instead of integrodifferential ones. The variational formulation reduces to two types of equations for total displacements and internal displacements (internal variables), namely to the equilibrium equation and the evolution equation for the internal displacements, which are fulfilled in the weak form. Using continuous trial functions, a continuous space-time finite element formulation is obtained with simultaneous discretization in space and time. Subdividing the total observation time interval into time slabs and introducing discontinuous trial functions, being continuous within time slabs and allowing jumps across interfaces, a more general discontinuous finite element formulation is obtained. The difference between these two formulations for one time slab consists in the satisfaction of initial conditions which are fulfilled exactly for the continuous formulation and in a weak form for the discontinuous case. The proposed approach has some very attractive advantages with respect to semidiscretization methods, regarding the possibility of adaptive space-time refinements and parallel processing on MIMD-parallel computers. The considered numerical examples show the effectiveness of simultaneous space-time finite element calculations and a high convergence rate for adaptive refinement. Numerical efficiency is an advantage of DGM in comparison with CGM for discontinuously changing (e.g. piecewise constant) boundary conditions in time.