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On the Stochastic Richards Equation Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften am Fachbereich Mathematik und Informatik der Freien Universit¨at Berlin On the Stochastic Richards Equation vorgelegt von Ralf Forster Berlin 2011 Betreuer: Prof. Dr. Ralf Kornhuber, Berlin Gutachter: Prof. Dr. Ralf Kornhuber, Berlin Prof. Dr. Omar Knio, Baltimore Tag der Disputation: 8. M¨arz 2011 Contents Introduction 1 1 Deterministic Richards equation in homogeneous soil 7 1.1 Basic tools in hydrology and Kirchhoff transformation . ...... 7 1.2 Boundary conditions and weak formulation . .. 12 2 The stochastic Richards equation 16 2.1 Randomfields............................... 16 2.2 Tensorspaces............................... 18 2.3 Weakformulation............................. 22 3 Discretization 29 3.1 TheKarhunen–Lo`eveexpansion. 29 3.1.1 Definition ............................. 30 3.1.2 Independent KL expansions and reformulation . 36 3.1.3 Computationalaspects. 42 3.2 Time discretization and convex minimization . .... 44 3.2.1 Timediscretization. 44 3.2.2 Formulation as a convex minimization problem . 46 3.2.3 Variationalinclusions . 51 3.3 Polynomial chaos and finite elements . 57 3.3.1 (Generalized)Polynomialchaos . 58 3.3.2 Finiteelementsinspace . 63 3.3.3 Approximation of the nonlinear functional . .. 65 3.3.4 Aconvergenceresultforthelimitcases . 71 3.4 Numericalexperiments. 78 3.4.1 Resultsforonespacedimension. 79 3.4.2 Resultsfortwospacedimensions . 81 4 Numerical solution 85 4.1 CommonmethodsforSPDEs . 85 4.1.1 StochasticGalerkinmethod . 86 4.1.2 Stochasticcollocation . 90 4.1.3 MonteCarlomethod. 92 4.2 Nonlinear Block Gauß–Seidel and minimization . .... 95 4.2.1 NonlinearBlockGauß–Seidel . 95 4.2.2 TransformationofthetensorproductPC . 99 4.2.3 Blockminimization. 106 4.3 Monotonemultigridmethod. 107 4.3.1 Multilevelcorrections . 107 4.3.2 Constrained Newton linearization with local damping . 109 4.3.3 Numericalresults. 115 4.4 Post-processing .............................. 117 5 A hydrological example 120 5.1 Linesmoothersforananisotropicgrid . 120 5.2 SolutionoftheRichardsequation. 123 Appendix A Exponential covariance 131 B Orthogonal polynomials 133 B.1 Hermitepolynomials . 133 B.2 Legendrepolynomials . 134 B.3 Jacobipolynomials .. .. .. .. .. .. .. .. .. .. .. .. 135 B.4 Laguerrepolynomials. 136 C Gaussian quadrature 137 List of Symbols 139 Bibliography 143 Zusammenfassung 151 Introduction Numerical simulation of complex systems is a rapidly developing field in mathemat- ical research with a growing number of applications in engineering, environmental sciences, medicine, and many other areas. This can be traced back to the increasing computer power on the one hand and more and more refined numerical techniques on the other hand. The main goal of these simulations is to make predictions which could reduce the costs of experiments or could be the basis for scientific or political decisions. In light of these consequences, the question of how reliable the obtained results are is of great importance. Since all simulations are, in the best case, only approximations to reality, the occurrence of errors is inevitably in some degree. As a first step, let us classify different types of errors which appear necessarily in the simulation of complex systems (according to [72]). First, we detect model errors, which are rooted in the description of observed real-life processes in mathematical terms. In this stage, some terms which only contribute to a negligible extent to the system are omitted. Moreover, one is often forced to impose certain assumptions to simplify the problem in order to be able to solve it, e.g. if one reduces the dimension of a problem or assumes that certain terms are constant. Secondly, in order to compute the simulation, one has to discretize the system and to solve it numerically, which leads to numerical errors. These errors are also in- evitably due to the finite representation of numbers in computers. This concerns the discretization itself and also the utilization of solvers, e.g. iterative solvers which only provide a solution up to a certain tolerance. Time and computer system re- strictions can limit the possibilities to pass to more and more refined discretizations or to solve up to maximal accuracy, even if it were possible theoretically. Thirdly, we encounter data errors, which can concern model parameters, boundary conditions or even the geometry of the system. Often, these data are unknown or only estimated roughly since exact measurements are impossible or too costly. In many cases, a perfect knowledge of the data is not possible due to their natural vari- ability. Eventually, the measurements themselves can be incorrect, e.g. by reason of external effects. The first source of errors is beyond the scope considered here. Regarding the second point, there has been great advantages in the numerical mathematics to control and reduce the error arising from discretization and numerical solution over the last decades. The third error originating from uncertainties in the input data and model parameters, however, has received less attention for a long time. This has changed since the pioneering work of Ghanem and Spanos [47] about stochastic finite elements and polynomial chaos, who presented a possibility to determine the response surfaces of the solution from the stochastic modeling of the uncertain input data. This method proved to be successful for many interesting applications 1 in physics and engineering (see, e.g., [24] for meteorology, [40, 46, 92] for hydrology, [86] for chemistry, and in particular the recent monograph of Le Maˆıtre and Knio [72] with applications in fluid dynamics). The idea of polynomial chaos consists in extending the solution space by further stochastic coordinates which come from the modeling of the uncertain input data and in discretizing the new space by global polynomial basis functions, while the time and space discretization for the other coordinates can be performed in the usual manner. In this way, this approach can be seen as an extension of deterministic finite element methods for partial differential equations. At the same time, it is in most cases more efficient and more accurate than other methods like, e.g., Monte Carlo or first-order methods (see the overview over alternative methods in [47] and [101]). There have been many improvements to the original idea from Ghanem and Spanos during the last twenty years, in particular the extension to generalized polynomial chaos [114], to multi-element polynomial chaos [107], to stochastic collocation [13], and to sparse finite element schemes [20]. Apart from the variety of applications as mentioned above, the analysis of this method in a theoretical way, however, mostly concentrated on investigating the diffusion equation with a stochastic diffusion as a prototype of an elliptic partial differential equation (e.g. [11, 29, 87, 103]). To the knowledge of the author, there is in particular no application of this method to variational inequalities. The focus of this thesis is the application of polynomial chaos methods to the Richards equation modeling groundwater flow in order to quantify the uncertainty arising from certain stochastic model parameters. By means of the Richards equa- tion, one can model water flows in saturated and in unsaturated porous media, where it is parabolic in the unsaturated and elliptic in the saturated regime. The main difficulty consists in the nonlinearities, since the saturation and the hydraulic conductivity are both dependent on the pressure. Moreover, the saturation appears in the time derivative, while the hydraulic conductivity is a factor in the spatial derivative. Berninger [18] recently presented a method to solve this problem without any linearization techniques by applying a Kirchhoff transformation and an implicit– explicit time discretization to obtain an elliptic variational inequality of second kind (see [48, Chapter 1] for the terminology) with a convex, lower semicontinu- ous functional. This variational inequality can be seen equivalently as a convex minimization problem, which in turn allows the proof of existence results for the solution and of convergence results for an appropriate finite element discretization. Moreover, monotone multigrid methods [68] providing fast and reliable solutions of the system are applicable and, since no linearization is involved, turn out to be robust with respect to varying soil parameters. Let us now consider the case that some parameters in the Richards equation are un- certain and consequently modeled as random functions, e.g. the permeability of the soil, the boundary conditions or source functions. These random functions should have a positive correlation length, which means that stochastic partial differential equations (SPDEs) with white noise like in [58] are not considered here. Our ulti- mate goal is to quantify the uncertainty of the solution of the Richards equation, viz. of the water pressure, in dependence of the uncertainty of the input parameters such that we are able to answer the following questions: (Q1) What is the average pressure in the domain, and how is it developing in time? 2 (Q2) In which areas of the soil do we encounter the largest uncertainty? (Q3) What is the probability that the soil is saturated after one minute/ one hour/one day at special points in the domain? (Q4) What is the probability that the pressure exceeds a fixed threshold value within a predetermined time? To achieve this goal, we first perform a spectral decomposition
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