Fictitious domains with separable preconditioners versus unstructured adapted meshes

Fictitious domain method with penalty for an incompressible fluid

“…Kobelkov proposed an iterative algorithm [5,6] to solve this weak form of the Possion equation which we extend to Navier-Stokes equations (3.2)-(3.6).…”

Fictitious Domain Methods for the Numerical Solution of Two-Dimensional Scattering Problems

“…The signs in the corner condition are chosen such that the derivatives are directed outward. The exact nonreflecting boundary condition on a circular boundary is given by 6) where M denotes the Dirichlet-to-Neumann (DtN) mapping, which relates the Dirichlet and Neumann data of this particular problem [23,24]. The summation symbol with a prime means that the first term in the sum is divided by two.…”

A moving mesh fictitious domain approach for shape optimization problems

“…From Assumption 3.1, Theorem 4.1 and the results in [28,39] it follows that the eigenvalues of the generalized eigenvalue problem 4 ], where the constants η 1 ≤ η 2 < 0 < η 3 ≤ η 4 are independent of the mesh step size h and the domain Ω α ∈ O. This guarantees that the number of required PMINRES iterations to reduce the norm of the residual by a prescribed factor is bounded from above with a constant which is independent of h and Ω α .…”

“…Now, the triangulation for the domain Ω is given by T Ω = {ψ Ω,τ (τ )} τ ∈TP . These mappings can be constructed, for example, using an explicit formula, the Laplacian smoothing [14] or combining either of these with a local fitting procedure as in [4,5]. An example of a triangulation T Ω for a circle obtained using the function ψ Ω constructed in Section 5.1 is shown in Figure 1 (b).…”

“…In this case, the formulation is typically done at the matrix level. See, for example, [1,4,6,34,39]. With the algebraic approach a lot of attention is given to the efficient solution of the arising linear problem.…”