In this thesis, we address the solution of time-dependent partial differential equations using the finite difference method and the multigrid method. The fundamental ideas of these approaches are illustrated through the solution of the Poisson equation. We also examine how to establish convergence of the multigrid method and determine the conditions under which it holds. The convergence rate is presented for different types of multigrid cycles. This knowledge is subsequently extended and applied to the solution of more complex, time-dependent problems, in particular the heat equation.
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