Numerical Algorithms for Algebraic Stabilizations of Scalar Convection-Dominated Problems

In dieser Arbeit wurden Finite-Elemente-Verfahren mit algebraischer Fluss\-kor\-rek\-tur (AFC) f\"ur station\"are Konvektions-Diffusions-Reaktions Gleichungen untersucht. Die beiden Hauptaspekte, die studiert wurden, sind iterative L\"oser f\"ur die auftretenden nichtlinearen Gleichungen und adaptive Gitterverfeinerung basierend auf a posteriori Fehlersch\"atzern. Die wichtigsten Ergebnisse der Arbeit sind im Folgenden zusammengefasst. Zun\"achst wurden Studien zu den L\"osern vorgestellt. Es wurden mehrere iterative L\"oser untersucht, darunter Fixpunktans\"atze und Methoden vom Newton-Typ. Die Newton Methoden reduzierten die Anzahl der Iterationen f\"ur bestimmte Beispiele, aber sie waren ineffizient bez\"uglich der Rechenzeit. Der einfachste Fixpunktansatz, n\"amlich \fpr, war auf Grund seiner Matrixeigenschaften am effizientesten. Algorithmische Komponenten, wie die Anderson-Beschleunigung, reduzierten die Anzahl der Iterationen in einigen Beispielen, aber sie lieferte keine Ergebnisse f\"ur den BJK-Limiter. In drei Dimensionen wurde ein iterativer L\"oser f\"ur feinere Gitter ben\"otigt, aber auch hier war \fpr die effizienteste Herangehensweise. Unabh\"angig von der Dimension war es einfacher, die Probleme mit dem Kuzmin-Limiter als mit dem BJK-Limiter zu l\"osen. Der zweite Hauptaspekt sind Studien zur a posteriori Fehlersch\"atzung. Es wurden zwei Ans\"atze zur Bestimmung einer oberen Schranke in der Energie\-norm untersucht, ein auf Resi\-duen basierender Ansatz (\emph{AFC-Energie} Technik) und ein anderer mit der SUPG-L\"osung (\emph{AFC-SUPG-Energie} Technik). Beide Techniken liefern keine robusten Sch\"atzungen bez\"uglich $\varepsilon$, aber es zeigte sich, dass der \emph{AFC-SUPG Energie} Ansatz einen besseren Effektivit\"ats\-index besa{\ss}. F\"ur den BJK-Limiter war die Effektivit\"at besser als f\"ur den Kuzmin-Limiter mit dem \emph{AFC-Energie} Ansatz, w\"ahrend beim \emph{AFC-SUPG Energie} Ansatz die Wahl des Limiters keine Rolle spielte. Im Zuge der adaptiven Gitterverfeinerung kann das Problem lokal diffusions-dominant werden. In diesem Falle muss man den BJK-Limiter verwenden, da man beim Kuzmin-Limiter eine reduzierte Konvergenzordnung beobachten kann. Im Hinblick auf die adaptive Gitterverfeinerung wurden Grenzschichten unterschiedlichen Typs besser mit dem \emph{AFC-Energie} Ansatz verfeinert als mit dem \emph{AFC-SUPG Energie} Ansatz. Schlie{\ss}lich wurden die Ergebnisse f\"ur die a posteriori Fehlersch{\"a}tzung auf Gitter mit h{\"a}ngenden Knoten angewandt. Zun\"achst wurden Ergebnisse bez\"uglich h\"angender Knoten von Lagrange-Elementen niedriger Ordnung auf Elemente h\"oherer Ordnung erweitert. Es zeigte sich in numerischen Studien, dass der Kuzmin-Limiter auf Gittern mit h{\"a}ngenden Knoten dem DMP nicht gen\"ugt, w{\"a}hrend der BJK-Limiter Ergebnisse lieferte, die dem DMP entsprachen. Die Grenzschichten wurden auf konform abgeschlossenen Gittern wesentlich besser approximiert als auf Gittern mit h{\"a}ngenden Knoten. Insgesamt sollte man Gitter mit h{\"a}ngenden Knoten nicht f\"ur AFC Verfahren verwenden.This thesis studies the Algebraic Flux Correction (AFC) schemes for the steady-state convection-diffusion-reaction equations. The work is done on two major aspects of these schemes, namely the iterative solvers for the nonlinear equations and a posteriori error estimation. The major findings of the thesis are summarized below. First, studies concerning the solvers are presented. Several iterative solvers are studied including fixed-point approaches and Newton-type methods. Newton methods reduce the number of iterations for certain examples but it is computationally inefficient. The most simple fixed point approach, namely the fixed point right-hand side is the most efficient because of its matrix structure. Algorithmic components such as Anderson acceleration reduced the number of iterations in some examples but it failed to give results for the BJK limiter. In three dimensions, an iterative solver is needed for finer meshes but here also the fixed point right-hand side is the most efficient. Irrespective of the dimension, it is easier to solve the problem with the Kuzmin limiter as that of the BJK limiter. In conclusion, one might get fewer iterations, with advanced methods but the simple fixed-point approach with dynamic damping is the most efficient in both dimensions. Second, studies for a posteriori error estimation is presented. Two approaches for finding the upper bound are investigated in the energy norm, one residual-based (AFC-Energy technique), and others using the SUPG solution (AFC-SUPG Energy technique). The AFC-Energy estimator is shown not to be robust with respect to $\varepsilon$ and hence, the AFC-SUPG approach gave a better effectivity index. For the BJK limiter, the effectivity is better than the Kuzmin limiter with the AFC-Energy approach, whereas for the AFC-SUPG approach the choice of limiter did not play a role. With adaptive grid refinement, the problem could become locally diffusion dominated and hence one has to use the BJK limiter as one can observe reduced order of convergence for the Kuzmin limiter. In regards to adaptive grid refinement, the AFC-Energy approach approximated the layer much better as compared to the AFC-SUPG approach. Lastly, the results for a posteriori error estimation are extended to grids with hanging nodes. First, results regarding hanging nodes are extended from lower-order Lagrange elements to higher-order elements. It was shown that the Kuzmin limiter fails to satisfy DMP on grids with hanging nodes, whereas the BJK limiter satisfies the DMP. The layers are properly approximated on conformally closed grids in comparison to grids with hanging nodes. Altogether, one should not use grids with hanging nodes for AFC schemes

Adaptive time-stepping for incompressible flow. Part II: Navier-Stokes equations

We outline a new class of robust and efficient methods for solving the Navier- Stokes equations. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the potential of our approach. © 2010 Society for Industrial and Applied Mathematics

On Iterative Algorithms for Quantitative Photoacoustic Tomography in the Radiative Transport Regime

In this paper, we describe the numerical reconstruction method for quantitative photoacoustic tomography (QPAT) based on the radiative transfer equation (RTE), which models light propagation more accurately than diffusion approximation (DA). We investigate the reconstruction of absorption coefficient and/or scattering coefficient of biological tissues. Given the scattering coefficient, an improved fixed-point iterative method is proposed to retrieve the absorption coefficient for its cheap computational cost. And we prove the convergence. To retrieve two coefficients simultaneously, Barzilai-Borwein (BB) method is applied. Since the reconstruction of optical coefficients involves the solution of original and adjoint RTEs in the framework of optimization, an efficient solver with high accuracy is improved from~\cite{Gao}. Simulation experiments illustrate that the improved fixed-point iterative method and the BB method are the comparative methods for QPAT in two cases.Comment: 21 pages, 44 figure

Domain Decomposition Based Hybrid Methods of Finite Element and Finite Difference and Applications in Biomolecule Simulations

The dielectric continuum models, such as Poisson Boltzmann equation (PBE), size modified PBE (SMPBE), and nonlocal modified PBE (NMPBE), are important models in predicting the electrostatics of a biomolecule in an ionic solvent. To solve these dielectric continuum models efficiently, in this dissertation, new finite element and finite difference hybrid methods are constructed by Schwartz domain decomposition techniques based on a special seven-box partition of a cubic domain. As one important part of these methods, a finite difference optimal solver --- the preconditioned conjugate gradient method using a multigrid V-cycle preconditioner --- is described in details and proved to have a convergence rate independent of mesh size in solving a symmetric positive definite linear system. These new hybrid algorithms are programmed in Fortran, C, and Python based on the efficient finite element library DOLFIN from the FEniCS project, and are well validated by test models with known analytical solutions. Comparison numerical tests between the new hybrid solvers and the corresponding finite element solvers are done to show the improvement in efficiency. Finally, as applications, solvation free energy and binding free energy calculations are done and then compared to the experiment data

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify the required regularity of the solution with respect to the random domain mapping for the use of multilevel quadrature, derive the coupling formulation, and show by numerical results that the approach is feasible

Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization

Topology optimization problems generally support multiple local minima, and real-world applications are typically three-dimensional. In previous work [I. P. A. Papadopoulos, P. E. Farrell, and T. M. Surowiec, Computing multiple solutions of topology optimization problems, SIAM Journal on Scientific Computing, (2021)], the authors developed the deflated barrier method, an algorithm that can systematically compute multiple solutions of topology optimization problems. In this work we develop preconditioners for the linear systems arising in the application of this method to Stokes flow, making it practical for use in three dimensions. In particular, we develop a nested block preconditioning approach which reduces the linear systems to solving two symmetric positive-definite matrices and an augmented momentum block. An augmented Lagrangian term is used to control the innermost Schur complement and we apply a geometric multigrid method with a kernel-capturing relaxation method for the augmented momentum block. We present multiple solutions in three-dimensional examples computed using the proposed iterative solver