Stability estimates for hybrid coupled domain decomposition methods
The monograph provides many useful stability estimates for constructing hybrid coupled domain decomposition (DD) discretization schemes on the basis of stable approximation of the Steklov-Poincaré operator. The Steklov-Poincaré operator plays an important role in non-overlapping DD methods with matching and non-matching grids. The book contains 5 chapters. Chapter 1 provides the necessary material on Sobolev spaces, saddle point problems, finite element spaces, projection operators, and quasi interpolant operators. The core results on stability of certain \(L_2\) projections onto different finite element spaces in some scale of Sobolev norms are derived in Chapter 2. The stability results are shown for finite element spaces on non-structured and non-uniform meshes, as they typically arise in adaptive computations, as well as for higher order finite element spaces. The Dirichlet-to-Neumann operator mapping the Dirichlet boundary data \(\gamma_0 u\) to the Neumann boundary data \(\gamma_1 u\) is given by the equation \[ \gamma_1 u(x)= S\gamma_0 u(x)- Nf(x),\quad x\in\Gamma= \partial\Omega,\tag{1} \] with the Steklov-Poincaré operator \(S\) and the Newton potential \(Nf\) belonging to some inhomogeneous partial differential equation \(L(x)u(x)= f(x)\), \(x\in\Omega\). Both \(S\) and \(Nf\) are only given implicitly. In Chapter 3, computable and stable approximations \(\widetilde S\) and \(\widetilde N\) to \(S\) and \(N\) are given on the basis of finite element as well as boundary element approximations. In Chapter 4 and 5, these approximations are used to derive stable discretization schemes for mixed boundary value problems and stable hybrid coupled non-overlapping domain decomposition equations. The numerical results presented for both cases confirm the theoretically predicted estimates. The reviewer recommends this monograph especially to people working in DD methods, in partial differential equation and boundary element methods.
scientific article; zbMATH DE number 1855774