Open Access

Andreas Gantner

• In life sciences, during the past few years increasing efforts have been devoted to the development of so-called "labs-on-a-chip". By definition, these are chip-based miniature laboratories that can be controlled electronically (and thus by computers). In spite of their smallness, the chip laboratories are able to conduct complex tasks in biology and chemistry on a few square centimeters where otherwise full-size laboratories are needed. The physicist Prof. Dr. Achim Wixforth from the University of Augsburg has developed a technology that brings about precise electronic control of chemical reactions on the surface of a biochip by utilizing surface acoustic waves. These SAWs can be easily excited by applying radiofrequency pulses to so-called interdigital transducers (IDTs) located on the surface of a piezoelectric substrate. As they move along, they are able to transport fluids and solid matters across the chip surface. Precise mathematical modeling and numerical simulations can helpIn life sciences, during the past few years increasing efforts have been devoted to the development of so-called "labs-on-a-chip". By definition, these are chip-based miniature laboratories that can be controlled electronically (and thus by computers). In spite of their smallness, the chip laboratories are able to conduct complex tasks in biology and chemistry on a few square centimeters where otherwise full-size laboratories are needed. The physicist Prof. Dr. Achim Wixforth from the University of Augsburg has developed a technology that brings about precise electronic control of chemical reactions on the surface of a biochip by utilizing surface acoustic waves. These SAWs can be easily excited by applying radiofrequency pulses to so-called interdigital transducers (IDTs) located on the surface of a piezoelectric substrate. As they move along, they are able to transport fluids and solid matters across the chip surface. Precise mathematical modeling and numerical simulations can help to achieve a precise understanding of the processes on a biochip. In this thesis, we develop a mathematical model for the SAW biochip based on the linearized equations of piezoelectricity. Here, the symmetry properties and energetic behavior of the piezoelectric material prove useful in the analysis of the solution behavior. Analytical studies of equations second order in time exist for a long time. However, the theories developed there cannot be applied directly: Only after a Gaussian elimination step resulting in the reduced Schur complement system the results from these books can be used. This requires an intrinsic study of the properties of the Schur complement operator. This operator also plays the all-dominant role in the analysis of the time harmonic approach. We will use the Riesz-Schauder theory to derive conditions on the solvability of this equation, a technique that is well-established e.g. for the Helmholtz equation. The study of the time-independent equations is of utmost importance for the analysis of time-harmonic and fully time-dependant problems. These equations now take the form of saddle point problems. Finite element problems where the lower diagonal matrix part $C$ satisfies $C = 0$ were extensively studied in the literature. Here, we focus on the case $C \not= 0$. Such problem have been studied in the context of stabilized mixed finite element methods or in the finite element modeling of slightly compressible fluids and solids. Also, some interior point methods in optimization result in systems with $C \not= 0$. In all of these cases the matrix $C$ has small norm compared to the other blocks. We establish a condition number bound for the Schur complement matrix that is independent of the meshsize parameter in the norm of the underlying continuous Sobolev spaces. Note that in our case the norm of $C$ is not small. The first iterative scheme for the solution of saddle point problems of a rather general type was that developed by Uzawa. Uzawa type methods are stationary schemes consisting of simultaneous iterations for the mechanical displacement $\b u$ and electric potential $\Phi$. Krylov subspace methods have turned out to be a strong alternative for the iterative solution of saddle point problems, particularly when a specially adapted preconditioner is at hand. For elliptic equations, construction principles for preconditioners are well-established, the fastest among them often resulting from domain decomposition and multilevel methods. For indefinite systems we can fall back on these construction principles and adapt the preconditioners to the indefinite setting. We show that the condition number of the thus obtained Schur complement matrix is bounded by a constant independent of the meshsize parameter.…