Least-squares finite element methods
Part I with 2 chapters provides a brief review of the finite element method and an introduction into the least-squares method. The latter can deal only with algebraic properties of the method, since the method associates to the abstract equation \(Lu=f\) with \(L:X \to Y\) the least-squares problem \[ \|Lu-f\|_0^2 \to \min! \] and the (Ritz-Galerkin) characterization of the minimum corresponds to the equation \(L^*Lu=L^*f\). Thus the condition number is squared, but this is not wanted. Therefore, Part II discusses concepts, how appropriate least-squares formulations for partial differential equations are obtained. It is easily described for the Poisson equation \(\Delta\phi = f\) which is rewritten as a system of first order: \(\nabla\phi = v\) and \(\text{div } v = f\). Associate to the system the problem \[ \|\nabla\phi - v \|_0^2 + \|\text{div } v - f\|_s^2 \to \min! \] where the norms refer to the Sobolev norms with the associated index. The discretization of the least-squares problem by finite elements yields only the optimal order of the error if the index \(s\) above is chosen such that system of partial differential equation induces an isomorphism between the Sobolev spaces. If this is true, one achieves \textit{energy balance} in terms of the book. Chapter 4 in Part II and Part III describe how the Sobolev spaces with energy balance are determined from the Agmon-Douglis-Nirenberg setting. We note that the theory by Babuška and Brezzi is applied in the finite element method for similar reasons. The most practical choice for the Poisson equation is obviously \(s=0\), but the energy balance requires \(s=-1\), i.e., a negative norm. Fortunately, the negative norm can be replaced by mesh-dependent norms, and it is elucidated later by numerical results that the mesh-dependent weights are essential. In Part III also the least-squares methods for several formulations of diffusion problems, electromagnetics and the Stokes problem are included. Some of them are probably new. Part IV is concerned with least-squares finite element methods for further problems: the Navier-Stokes equations, parabolic and hyperbolic partial differential equations, and control problems. A further chapter provides a brief discussion of various problems as, e.g., advection-diffusion problems, higher order elements, domain decomposition, and multi-physics problems. Part V with supplementary material does not only contain the usual information on Sobolev spaces. It provides a brief discussion of the Agmon-Douglis-Nirenberg theory as far as it is necessary for the application to least-squares methods. The DeRham complex with the Sobolev spaces \(H^1\), \(H(\text{curl})\), \(H(\text{div})\), and \(L_2\) is listed as well as the discrete analogues. Recent results that are incorporated in the book, refer in particular to negative norm least-squares finite element methods. Note that least-squares formulations refer to the target spaces of the (systems of) differential operators, and minimal regularity leads often to negative norms. In contrast to this fact, the finite element method and the bilinear forms there refer to the domain of the differential operators, and negative norms enter usually only if Lagrange multipliers are involved. The reader will profit from the modern representation of least-squares methods that does not stop when negative norms or the DeRham complex come into the play.
scientific article; zbMATH DE number 5292600