With partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is ( w i , ∂ ∂ u j ∂ ∂ t ) = − − ( w i , ( u ⋅ ⋅ ∇ ∇ ) u j ) − − ν ν ( ∇ ∇ w i : ∇ ∇ u j ) + ( w i , f S ) .