A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier-Stokes equations

Two‐grid stabilized algorithms for the steady Navier–Stokes equations with damping

“…The velocity-pressure finite element spaces are characterized by a uniform triangulation T 𝜇 (Ω) (here, 𝜇 = h or H) with the quadratic equal-order finite elements pair. In our numerical tests, an attractive feature is that the projection (Π 𝜇 ∇p 𝜇 , ∇q) is computed by using the first-order Gauss integration ∫ K,1 ∇p 𝜇 • ∇qdx over the element K, avoiding the construction of the projection operator Π 𝜇 (noting that, ∫ K,1 ∇p 𝜇 • ∇qdx is equivalent to (Π 𝜇 ∇p 𝜇 , ∇q) as stated in Zheng and Shang 38 ). Besides, to ensure optimal order convergence rates of our proposed algorithms, the mesh sizes between h and H are related below.…”

“…The velocity‐pressure finite element spaces are characterized by a uniform triangulation $$$ {T}_{\mu}\left(\Omega \right) $$$ (here, $$$ \mu =h $$$ or $$$ H $$$) with the quadratic equal‐order finite elements pair. In our numerical tests, an attractive feature is that the projection $$$ \left({\Pi}_{\mu}\nabla {p}_{\mu },\nabla q\right) $$$ is computed by using the first‐order Gauss integration $$$ {\int}_{K,1}\nabla {p}_{\mu}\cdotp \nabla qdx $$$ over the element $$$ K $$$, avoiding the construction of the projection operator $$$ {\Pi}_{\mu } $$$ (noting that, $$$ {\int}_{K,1}\nabla {p}_{\mu}\cdotp \nabla qdx $$$ is equivalent to $$$ \left({\Pi}_{\mu}\nabla {p}_{\mu },\nabla q\right) $$$ as stated in Zheng and Shang 38 ).…”

“…Recently, many investigations have focused on stabilization of the lowest equal-order elements pair using the projection of the pressure onto the piecewise constant space (see, e.g., other works [27][28][29][30][31][32][33] ) and of the quadratic equal-order elements pair using the projection of the gradient of pressure onto the piecewise constant space (cf. other studies [34][35][36][37][38] ). Most attractive features of these stabilized methods are free of parameter, avoiding higher-order derivatives or edge-based data structures, and stabilization being completely local at the element level.…”

A Parallel Finite Element Algorithm for the Unsteady Oseen Equations

“…In the last decades, based on the idea of Xu and Zhou [1][2][3] for local and parallel finite element discretizations, some new local and parallel algorithms have been proposed for eigenvalue problems [3][4][5], the stationary incompressible magnetohydro dynamicsthe [6], the steady Stokes equations [7][8][9][10], the Stokes/Darcy problem [11], the steady Navier-Stokes equations [12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the stream function form of Navier-Stokes equations [26]. These algorithms have less communication complexity than current standard approaches and allow existing sequential PDE codes to run in a parallel environment without a large investment in recoding.…”