By　investigating　iterative　methods　for　a　constrained　linear　model,　we　propose　a　new　class　of　fully　connected　V-cycle　MgNet　for　long-term　time　series　forecasting,　which　is　one　of　the　most　difficult　tasks　in　forecasting.　MgNet　is　a　CNN　model　that　was　proposed　for　image　classification　based　on　the　multigrid　(MG)　methods　for　solving　discretized　partial　differential　equations　(PDEs).　We　replace　the　convolutional　operations　with　fully　connected　operations　in　the　existing　MgNet　and　then　apply　them　to　forecasting　problems.　Motivated　by　the　V-cycle　structure　in　MG,　we　further　propose　the　FV-MgNet,　a　V-cycle　version　of　the　fully　connected　MgNet,　to　extract　features　hierarchically.　By　evaluating　the　performance　of　FV-MgNet　on　popular　datasets　and　comparing　it　with　state-of-the-art　models,　we　show　that　the　FV-MgNet　achieves　better　results　with　less　memory　usage　and　faster　inference　speed.　In　addition,　we　develop　ablation　experiments　to　demonstrate　that　the　structure　FV-MgNet　is　the　best　choice　among　the　many　variants.