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In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite difference approximation in space while the semi-discrete problem is discretized in time using the  ϑ -scheme. The simultaneous treatment of all time instances leads to a global system of linear equations which provides the potential for a higher degree of parallelization of multigrid solvers due to the increased number of degrees of freedom per spatial unknown.

Fourier analysis of a time-simultaneous two-grid algorithm using a damped Jacobi waveform relaxation smoother for the one-dimensional heat equation