Importance of parameter optimization in a nonlinear stabilized method adding a crosswind diffusion
The problem under consideration is the following \[ -\varepsilon\Delta u+\mathbf{b}\cdot u+cu=f \text{ in }\Omega,\quad u=u_b \text{ on } \Gamma^D,\quad \varepsilon\frac{\partial u}{\partial n}=g\text{ on } \Gamma^N, \] where \(\Omega\subset\mathbb R^d\), \(d=2,3\) is a bounded domain with Lipschitz polyhedral boundary \(\partial\Omega\), \(\Gamma^D\) and \(\Gamma^N\) are relatively open subsets of \(\partial\Omega\), \(\Gamma^D\cap\Gamma^N=\emptyset\), and \(\Gamma^D\cup\Gamma^N=\partial\Omega\). Furthermore, \(\varepsilon>0\) is a constant diffusivity, \(\mathbf{b}\in W^{1,\infty}(\Omega)^d\) is a convective field, \(c\in L^{\infty}(\Omega)\) is a reaction coefficient, \(f\in L^2(\Omega)\) is an outer source of \(u\), \(u_b\in H^{1/2}(\Gamma^D)\) is the Dirichlet boundary condition, and \(g\in L^2(\Gamma^N)\) is a function prescribing the Neumann boundary condition and some additional assumptions. The authors introduce a finite element space and use so called streamline upwind/Petrov-Galerkin (SUPG) method [\textit{A. N. Brooks} and \textit{T. J. R. Hughes}, Comput. Methods Appl. Mech. Eng. 32, 199--259 (1982; Zbl 0497.76041)] for approximation. They describe a certain iteration procedure which is called SOLD method and assert that `` this approach can lead to more physically meaningful solutions than techniques considered before''. Numerical experiments are given.
scientific article; zbMATH DE number 7337491
scientific article