Optimal first-order error estimates of a fully segregated scheme for the Navier–Stokes equations

In this article, a first‐order linear fully discrete pressure segregation scheme is studied for the time‐dependent incompressible magnetohydrodynamics (MHD) equations in three‐dimensional bounded domain. Based on an incremental pressure projection method, this scheme allows us to decouple the MHD system into two sub‐problems at each time step, one is the velocity‐magnetic field system, the other is the pressure system. Firstly, a coupled linear elliptic system is solved for the velocity and the magnetic field. Next, a Poisson‐Neumann problem is treated for the pressure. We analyze the temporal error and the spatial error, respectively, and derive the temporal‐spatial error estimates of for the velocity and the magnetic field in the discrete space and for the pressure in the discrete space without imposing constraints on the mesh width and the time step size . Finally, some numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the method.

“…Lemma The following relations hold [21]:

\begin{array}{ll}\hfill & {\left\Vert {\tilde{\mathbf{e}}}_u^{n+1}\right\Vert}^2={\left\Vert {\mathbf{e}}_u^{n+1}\right\Vert}^2+{\left\Vert \Delta t\Big(\nabla \left({e}_p^{n+1}-{e}_p^n-\Delta t{\delta}_tp\left({t}_{n+1}\right)\right)\right\Vert}^2,\\ {}\hfill & {\left\Vert {\mathbf{e}}_u^{n+1}-{\tilde{\mathbf{e}}}_u^{n+1}\right\Vert}^2\le {\left\Vert {\tilde{\mathbf{e}}}_u^{n+1}-{\mathbf{e}}_u^n\right\Vert}^2,\\ {}\hfill & {\left\Vert {\mathbf{e}}_u^{n+1}\right\Vert}_{H^1}\le {\left\Vert {\tilde{\mathbf{e}}}_u^{n+1}\right\Vert}_{H^1}.\end{array}

…”

Section: Time Discrete Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unconditional optimal first‐order error estimates of a full pressure segregation scheme for the magnetohydrodynamics equations

Yang,

Jiang

2024

“…Remark This scheme, based on a so‐called incremental pressure projection method (see e.g., ), decouples the computation of

(ϕ^{n + 1}, μ^{n + 1}, {\overset{true˜}{u}}^{n + 1})

and p n + 1 . These projection methods are widely used because of their efficiency for large scale numerical simulations.…”

Section: The Time‐discretization Schemementioning

confidence: 99%

“…Following the idea of , the fully discrete finite element scheme reads: Given

(u_{h}^{n}, {\overset{true˜}{boldu}}_{h}^{n}, ϕ_{h}^{n}, p_{h}^{n}) \in ({boldV}_{h} + \nabla X_{h}) \times {boldV}_{h} \times X_{h} \times X_{h}^{0}

, find

({\overset{true˜}{boldu}}_{h}^{n + 1}, ϕ_{h}^{n + 1}, μ_{h}^{n + 1}, ℒ_{h}^{n + 1}) \in {boldV}_{h} \times X_{h} \times X_{h} \times M_{h}

satisfying:

\begin{array}{l} \frac{1}{δ italict} ((),, {\overset{u}{true}}_{h}^{n + 1}, v) + b ((),,, {\overset{u}{true}}_{h}^{n}, {\overset{u}{true}}_{h}^{n + 1}, v) + λ ((),, ϕ_{h}^{n} \nabla μ_{h}^{n + 1}, v) + a_{0} ((),, {\overset{u}{true}}_{h}^{n + 1}, v) - λ \int_{Γ} ℒ_{h}^{n + 1} \nabla_{τ} ϕ_{h}^{n} \cdot v_{τ} d Γ \\ = \frac{1}{δ italict} ((),, {boldu}_{h}^{n}, v) - ((),, \nabla p_{h}^{n}, v) + β \int_{Γ} Q_{h} u_{w} \cdot v_{τ} d Γ, \\ - ((),, μ_{h}^{n + 1}, η) - \int_{Γ} \end{array}

…”

Section: Fully Discrete Finite Element Schemementioning

confidence: 99%

Energy stable numerical scheme for the viscous Cahn–Hilliard–Navier–Stokes equations with moving contact line

Cherfils

Petcu

2019

In the present article we study the numerics of the viscous Cahn–Hilliard–Navier–Stokes model, endowed with dynamic boundary conditions which allow us to take into account the interaction between the fluids interface and the moving walls of the physical domain. In what follows, we propose an energy stable temporal scheme for the problem and we prove the stability and the unconditional solvability of the discretization proposed. We also propose a fully discrete scheme for which we prove the stability and the unconditional solvability. Numerical simulations are presented to illustrate the theoretical results.

“…This well‐known Chorin–Temam projection method is a two‐step scheme, computing firstly an intermediate velocity via a linear elliptic problem and secondly a velocity–pressure pair via a solenoidal (divergence‐free) L 2 ‐projection problem. For the variants of the projection methods, we can refer to [17–20, 24, 31, 39]. However, the drawback is that the end‐of‐step velocity does not satisfy the exact boundary conditions and the discrete pressure satisfies an “artificial” Neumann boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Error estimates of an operator‐splitting finite element method for the time‐dependent natural convection problem

Yang

Huang

Jiang

2022

In this article, an operator‐splitting (or a fractional‐step) finite element method is proposed for the numerical solution of the time‐dependent natural convection problem. We first analyze the time‐discrete fractional‐step scheme, which decouples the considered system into three subproblems, and each subproblem can be solved more easily than the original one. Under some mild regularity assumptions, the stability and error estimates for the velocity and the temperature are established rigorously. In addition, a full discrete fractional‐step scheme based on the time‐discrete scheme is proposed and the rigorous error analysis is presented. We derive the temporal–spatial error estimates of scriptO(normalΔt+h)$$ \mathcal{O}\left(\Delta t+h\right) $$ for the velocity and the temperature in the discrete space l2()H1∩l∞()L2$$ {l}^2\left({H}^1\right)\cap {l}^{\infty}\left({L}^2\right) $$ under the constraint Δt ≥ Ch. Numerical experiments are performed to confirm the theoretical predictions and demonstrate the efficiency of the method.