Parallel fictitious domain method for a non-linear elliptic Neumann boundary value problem (Q2760339)

A linear elliptic equation with Neumann boundary conditions and variable coefficients is given on a 2D domain \(\Omega \). It serves as a model problem to demonstrate the proposed method the basic idea of which comprises three steps. First, \(\Omega \) is extended to a rectangular domain \(\widehat \Omega \), where a uniform mesh \(\widehat \Pi _h\) consisting of right angle triangles is defined. A mesh \(\Pi _h\) on \(\Omega \) is introduced via local modifications of \(\widehat \Pi _h\) (to approximate \(\partial \Omega \)) and a restriction to \(\Omega \). Second, besides a system of linear algebraic equations \(Au=f\) resulting from a finite element discretization on \(\Pi _h\), a matrix \(B\) corresponding to the discretized Laplace operator on \(\widehat \Pi _h\) is considered. Next, the matrix \(A\) is augmented by zero blocks to a matrix \(C\) such that NEWLINE\[NEWLINE C\binom {u}{w}\equiv \begin{pmatrix} A&0\\0&0\end{pmatrix} \binom {u}{w}=\binom {f}{0}. \tag{1} NEWLINE\]NEWLINE System (1) is solved in the subspace \(V\equiv (\operatorname {Ker}(C))^\perp \) by the preconditioned conjugate gradient method. The pseudo-inverse \(B^+\) of \(B\) serves as a preconditioner. It is shown that \(C\) and \(B\) are spectrally equivalent on \(V\). NEWLINENEWLINENEWLINEThe method is parallelized, i.e., the extended domain \(\widehat \Omega \) is subdivided into rectangular subdomains with mutually balanced amount of nodes. NEWLINENEWLINENEWLINEFor numerical experiments, the authors chose a nonlinear problem solved by the Newton method. Namely, the full potential subsonic flow around an aerofoil. Tables and graphs give some evidence that both preconditioning and parallelization are efficient. It would be interesting, however, to have also a comparison with a method using only the mesh \(\Pi _h\).