On the accuracy of the finite element method plus time relaxation

Source abstract

If u ¯ \overline {u} denotes a local, spatial average of u u , then u ′ = u − u ¯ u^{\prime }=u-\overline {u} is the associated fluctuation. Consider a time relaxation term added to the usual finite element method. The simplest case for the model advection equation u t + a → ⋅ ∇ u = f ( x , t ) u_{t}+\overrightarrow {a}\cdot \nabla u=f(x,t) is $$( u h , t + a → ⋅ ∇ u h , v h ) + χ ( u h ′ , v h ′ ) = ( f ( x , t ) , v h ) . (u_{h,t}+\overrightarrow {a}\cdot \nabla u_{h},v_{h})+\chi (u_{h}^{\prime },v_{h}^{\prime })=(f(x,t),v_{h}).$$ We analyze the error in this and (more importantly) higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation.