Subgrid Stabilized Defect Correction Methods for the Navier–Stokes Equations

Abstract

We consider the synthesis of a recent subgrid stabilization method with defect correction methods. The combination is particularly efficient and combines the best algorithmic features of each. We prove convergence of the method for a fixed number of corrections as the mesh size goes to zero and derive parameter scalings from the analysis. We also present some numerical tests which both verify the theoretical predictions and illustrate the method's promise.

A Two-Level Defect–correction Method for Navier–stokes Equations

“…From the analysis provided in [9] and [5] we see that if the solution to (1.1) is H 2 -regular, for the two correction schemes above, only one correction step is necessary to yield an H 1 optimal order approximation when σ = O(ξ λ…”

“…In our TLDC method we fix M = 75 and let m (≤M) change between 3 and M. For SGM, OLDC and SDC, we compute the associated approximations with respect to different M (3 ≤ M ≤ 75). For the SDC in particular, we are careful to choose m to keep the relation m ∼ M 1/2 as the analysis in [5].…”

“…This will eventually lead to a divergence of the iterative procedure. At the time of writing, application of the defect-correction type methods appears to be becoming a key component; see, for example, Layton et al [2,5,7,9], Stetter [11] and Böhmer [3]. The defect-correction method is an iterative improvement technique which is well established for solving nonlinear steady-state problems.…”

“…Subsequently, Layton also provided some further studies based on the defect-correction method for NavierStokes equations (see [9] and the references therein). Recently, Kaya et al [5] considered the synthesis of a subgrid stabilization method with defect-correction methods for the steady-state Navier-Stokes equations. In this paper we want to combine the two-level strategy with the defect-correction method to solve the steady-state Navier-Stokes equations with high Reynolds number.…”

“…In this paper we want to combine the two-level strategy with the defect-correction method to solve the steady-state Navier-Stokes equations with high Reynolds number. Here, we point out that all the computations in the subgrid defect-correction scheme studied in [5], both the defect step and the correction step, are carried out on the fine-level subspace which is actually a one-level method in spite of two mesh scales. However, in our two-level defect-correction scheme presented in a subsequent section, we execute the defect step in the coarse-level subspace and only solve a linear equation in fine-level subspace, which greatly reduces the computational time.…”