Analysis und Stabilisierungsmethoden für singulär gestörte Probleme und Systeme
Zusammenfassung der Projektergebnisse
We considered the singularly perturbed Oseen equations that can be seen as a simplification of the Navier–Stokes equations in fluid dynamics. Even more simplified problems like convection-diffusion problems are widely analysed in literature. Since now the structure of the solutions of these more complicated problems was unknown. By using a corresponding stream function ψ on the unit square we could prove a solution decomposition of ψ into a smooth part and layer parts ψ(x, y) = S(x, y) + εH(x, y) exp{−b1 x/ε} + εI(x, y) exp{−b2 y/ε}, where the functions S, H, I and J can be bounded independently of the small perturbation parameter ε. As the velocity u of the Oseen equations can be written as u = curl ψ, we assume a similar decomposition for the unknowns of the original problem. This leads to the possibility to create layer-adapted S-type meshes for the solution (u, p) of the original Oseen equations. On these meshes we introduce a finite element method and stabilised version, where a grad-div stabilisation term (γ div u, γ div v) is added to the bilinear form. We consider approximations of the velocity by Qk -elements and of the pressure by Qk−1 elements for k ≥ 2, which gives the Taylor-Hood family of elements. Alternatively we also considered a discontinuous approximation of the pressure by Pk−1 -elements with k ≥ 2. Both pairs of elements are known on isotropic meshes to fulfil an inf-sup condition. On anisotropic meshes we were able to prove |||(u − uN , p − pN )||| ≤ C (1 + 1 / B(N, ε, Jk−1 p − pN)) EB(N, ε)^k, where the energy norm is given with a constant α > 0 by |||(v, q)|||2 = ε ||∇v||^2 L2 (Ω) + c0 ||v||^2 L2 (Ω) + ||γ div v||^2 L2 (Ω) +α ||q||^2 L2 (Ω). In the estimation EB(N, ε) = h + l + N^−1 max |ψ'| is a measure depending on the used S-type mesh. For the standard Shishkin mesh with piecewise equidistant meshes we have EB(N, ε) ≤ CN −1 ln N while for the Bakhvalov–Shishkin mesh with additional grading the result improves to EB(N, ε) ≤ CN^−1 for ε small enough. Furthermore, our numerical experiments have shown that the special version of the inf-sup constant B(N, ε, J^k−1 p − pN ), which depends on the numerically computed pressure qN and an interpolation of the exact pressure p, can also be bounded by a constant independent of ε. Numerical experiments confirm the obtained uniform and optimal bound of the numerical method. The main goal of the project, “to investigate the solution behaviour of problems like the Oseen-equations together with suitable stabilisation methods”, has been tackled and a solution decomposition was found, as well as a uniform error analysis for Galerkin and grad-div stabilised FEM with optimal order convergence was made.
