This survey paper is based on three talks given by the second author at the London Mathematical

www.elsevier.com/locate/apnum A dimensional split preconditioner for Stokes and linearized

Abstract: Microfluidic biochips are devices that are designed for high throughput screening and hybridization in genomics, protein profiling in proteomics, and cell analysis in cytometry. They are used in clinical diagnostics, pharmaceutics and forensics. The biochips consist of a lithographically produced network of channels and reservoirs on top of a glass or plastic plate. The idea is to transport the injected DNA or protein probes in the amount of nanoliters along the network to a reservoir where the chemical analysis is performed. Conventional biochips use external pumps to generate the fluid flow within the network. A more precise control of the fluid flow can be achieved by piezoelectrically agitated surface acoustic waves (SAW) generated by interdigital transducers on top of the chip, traveling across the surface and entering the fluid filled channels. The fluid and SAW interaction can be described by a mathematical model which consists of a coupling of the piezoelectric equations and the compressible Navier-Stokes equations featuring processes that occur on vastly different time scales. In this contribution, we follow a homogenization approach in order to cope with the multiscale behavior of the coupled system that enables a separate treatment of the fast and slowly varying processes. The resulting model equations are the

Abstract. This paper is devoted to the numerical treatment of linear optimality systems (OS) arising in connection with inverse problems for partial differential equations. If such inverse problems are regularized by Tikhonov regularization, then it follows from standard theory that the associated OS is well-posed, provided that the regularization parameter α is positive and that the involved state equation satisfies suitable assumptions. We explain and analyze how certain mapping properties of the operators appearing in the OS can be employed to define efficient preconditioners for finite element (FE) approximations of such systems. The key feature of the scheme is that the numberof iterations needed to solve the preconditioned problem by the minimal residual method is bounded independentlyof the mesh parameter h, used in the FE discretization, and only increases moderately as α → 0. More specifically, if the stopping criterion for the iteration process is defined in terms of the associated energy norm, then the number of iterations required (in the severely illposed case) cannot grow faster than O((ln(α)) 2). Our analysis is based on a careful study of the involved operators which yields the distribution of the eigenvalues of the preconditioned OS. Finally, the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by Borzi, Kunisch and Kwak [14] and an inverse problem arising in connection with electrocardiography [41]. 1.