UMD Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/3
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Item Model Order Reduction for Parameter-Dependent Partial Differential Equations with Constraints(2023) Davie, Kayla Diann; Elman, Howard C; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Reduced-order methods have been used to reliably solve discrete parametrized mathematical models and to lessen the associated computational complexity of these models by projecting them onto spaces of reduced order. The reduced-order spaces are spanned by a finite number of full-order solutions, a reduced basis, that, if well-chosen, provide a good approximation of the entire solution manifold. Reduced-order methods have been used with various problem classes including different types of constrained parametrized problems such as constrained parametrized partial differential equations (PDEs) where the constraints are built into the PDEs, and parametrized PDE-constrained optimization problems, or PDE-control problems, where the constraints themselves are PDEs. In the deterministic setting, both of these problem classes involve discrete models that are of saddle-point form and can be computationally expensive to solve. It is well known that saddle-point problems must satisfy an inf-sup condition to ensure stability of the solution, thus, solving deterministic variations of these models requires the consideration and satisfaction of an inf-sup stability condition. When these models are subject to parametrization, the solution to a deterministic problem is sought for many parameter values. Reduced-order models for these problem classes are often constructed so that they mirror the full-order models and are also of saddle-point form. In the established RB methods we study for the problem classes explored in this thesis, the reduced basis is represented as a block-diagonal matrix that produces a saddle-point reduced system and is augmented to satisfy inf-sup stability. Two methods of building an augmented RB to ensure inf-sup stability that have been well studied are augmentation by aggregation and augmentation by the supremizer. We present a comparative study of these two common methods of stabilizing reduced order models, through use of the supremizer and through aggregation, and compare the accuracy, efficiency, and computational costs associated with them for solving the parametrized PDE-control problem. We propose a new approach to implementing the RB basis, the stacked reduced basis, that produces a reduced system that is not of saddle-point form. Implementing the stacked reduced basis avoids the necessity to satisfy the inf-sup condition to ensure stability and therefore, to augment the reduced bases spaces. This results in a reduced basis system of smaller order, which reduces the computational work in the online step. While inf-sup stability is avoided, there are still issues with the stability of the stacked reduced system during RB construction, particularly for the constrained PDE problem class. We show that this can be addressed by penalization and present results to show that penalization improves the stability of an established augmented RB method as well. We present numerical results to compare the new approach to two developed ways of implementing the RB method (with both commonly accepted choices of augmentation) and prove the efficiency of the proposed approach. We study the efficiency of the stacked reduced basis for both PDE-control problems and constrained PDEs subject to parametrization.Item A Class of Stable, Efficient Navier-Stokes Solvers(2006-05-10) Liu, Jie; Liu, Jian-Guo; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We study a class of numerical schemes for Navier-Stokes equations (NSE) or Stokes equations (SE) for incompressible fluids in a bounded domain with given boundary value of velocity. The incompressibility constraint and non-slip boundary condition have made this problem very challenging. Their treatment by finite element method leads to the well-known inf-sup compatibility condition. Their treatment by finite difference method leads to the very popular projection method, which suffers from low resolution near the boundary. In [LLP], the authors propose an unconstrained formulation of NSE or SE, which replace the divergence-free constraint by a pressure equation with an appropriate boundary condition. All of the schemes in this thesis are based on this new formulation. In contrast to traditional methods, these schemes do not need to fulfill the traditional inf-sup compatibility condition between velocity space and pressure space. More importantly, they can achieve high-order accuracy very easily and are very efficient due to the decoupling of the update of velocity and pressure. They can even be proved to be unconditionally stable. There are two ways to analyze the schemes that we propose. The first is based upon the sharp estimate of the pressure in [LLP]. The second relies on a nice identity. Using the pressure estimates, we propose and study a $C^1$ finite element (FE) scheme for the steady-state SE as well as for the time-dependent NSE. For steady-state SE, we can either use an iterative scheme or solve velocity and pressure together. Using the nice identity, we prove that the semi-discrete iterative scheme for the steady-state SE converges ("semi-discrete" means that the spatial variable are kept continuous). This identity will also be crucial for our proofs of the stability and error estimates of the time-dependent $C^0$ FE schemes. Associated numerical computations demonstrate stability and accuracy of these schemes. We also present the numerical results of yet another $C^0$ FE scheme ([JL]) for the time-dependent NSE for which the theory of the fully discrete case is yet lacking.