Multigrid convergence for nonsymmetric, indefinite variational problems and one smoothing step (Q1819888)

The convergence of the multigrid method in solving elliptic variational problems in the nonsymmetric and indefinite case is studied, i.e. a bilinear form \(a=a_ s+b\) is considered, where \(a_ s\) is a symmetric and positive definite part of \(a\). A scale of norms by this symmetric positive definite bilinear form is introduced: \(\|| \cdot \||_{\theta}\), \(\theta\in R\). A natural assumption on the approximation properties of a discrete subspace \({\mathcal S}\) of the considered Hilbert space \({\mathcal H}\) is imposed, i.e. A 1. There exists a constant \(\delta >0\) such that for some \(0<\alpha \leq 1\) the estimate \(\|| u-v\||^ 2_ 1\leq \delta^{\alpha}\|| u\||^ 2_{1+\alpha}\) for any \(u\in {\mathcal H}\) and some v from \({\mathcal S}\) (depending on u) holds true; A 2. The second assumption is that the spaces \({\mathcal S}\) and \({\mathcal T}:=^{def}\{u\in {\mathcal H}:\) \(a(u,v)=0\) for all \(v\in {\mathcal S}\}\) are almost orthogonal, i.e. \[ | b(u,v)| \leq \tau \|| u\||_ 1\|| v\||_ 1,\quad u\in {\mathcal T},\quad v\in {\mathcal H}, \] for some sufficiently small constant \(\tau <1\); and finally \[ A 3.\quad \forall \quad u,v\in {\mathcal H}\quad | b(u,v)| \leq \gamma \|| u\||_ 1\|| v\||_ 0 \] for some sufficiently small constant \(\gamma <1\). For the smoothing procedure the Richards iteration \[ u_ 1=u_ 0-\omega (Au_ 0-f),\quad 0<\omega <2, \] is used. (A is scaled in such a way that \(\rho (A_ s)=1\). \(A_ s\) corresponds to the symmetric positive definite part \(a_ s\) of a.) The convergence of the following two-level method is studied. Step 1. Given \(u_ 0\in {\mathcal H}\) perform a smoothing step \(u_ 1=u_ 0-\omega (Au_ 0-f);\) Step 2. Solve approximately the correction problem for unknown q with fixed \(u_ 1\) \(q\in {\mathcal S}:\) \(a(u_ 1+q,v)=g(v)\) for all \(v\in {\mathcal S}\), \(u_ 2=u_ 1+q;\) Step 3. Put \(u_ 0=u_ 2\) and go to step 1. The convergence factor is estimated in the \(\|| \cdot \||_ 1\) norm. The convergence of W and V-cycles of the multigrid iteration is studied. The assumptions are verified for a model second order elliptic problem.