164 research outputs found
Stationary Multiple Spots for Reaction-Diffusion Systems
In this paper, we review
analytical methods for a rigorous study of the
existence and stability of stationary, multiple
spots for reaction-diffusion systems. We will
consider two classes of reaction-diffusion
systems: activator-inhibitor systems (such as
the Gierer-Meinhardt system) and
activator-substrate systems (such as the
Gray-Scott system or the Schnakenberg model).
The main ideas are presented in the context of
the Schnakenberg model, and these results are
new to the literature.
We will consider the systems in a
two-dimensional, bounded and smooth domain for small diffusion
constant of the activator.
Existence of multi-spots is proved using tools
from nonlinear functional analysis such as
Liapunov-Schmidt reduction and fixed-point
theorems. The amplitudes and positions of spots
follow from this analysis.
Stability is shown in two parts, for
eigenvalues of order one and eigenvalues
converging to zero, respectively. Eigenvalues
of order one are studied by deriving their
leading-order asymptotic behavior and reducing
the eigenvalue problem to a nonlocal eigenvalue
problem (NLEP). A study of the NLEP reveals a
condition for the maximal number of stable
spots.
Eigenvalues converging to zero are investigated
using a projection similar to Liapunov-Schmidt
reduction and conditions on the positions for
stable spots are derived. The Green's function
of the Laplacian plays a central role in the
analysis.
The results are interpreted in the biological,
chemical and ecological contexts. They are
confirmed by numerical simulations
Numerical Treatment of Non-Linear singular pertubation problems
Magister Scientiae - MScThis thesis deals with the design and implementation of some novel numerical methods for non-linear singular pertubations problems (NSPPs). It provide a survey of asymptotic and numerical methods for some NSPPs in the past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information.South Afric
On Discontinuous Galerkin Methods for Singularly Perturbed and Incompressible Miscible Displacement Problems
This thesis is concerned with the numerical approximation of problems of fluid flow, in particular the stationary advection diffusion reaction equations and the time dependent, coupled equations of incompressible miscible displacement in a porous medium.
We begin by introducing the continuous discontinuous Galerkin method for the singularly perturbed advection diffusion reaction problem. This is a method which coincides with the continuous Galerkin method away from internal and boundary layers and with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method.
We then turn our attention to the equations of incompressible miscible displacement for the concentration, pressure and velocity of one fluid in a porous medium being displaced by another. We show a reliable a posteriori error estimator for the time dependent, coupled equations in the case where the solution has sufficient regularity and the velocity is bounded. We remark that these conditions may not be attained in physically realistic geometries. We therefore present an abstract approach to the stationary problem of miscible displacement and investigate an a posteriori error estimator using weighted spaces that relies on lower regularity requirements for the true solution.
We then return to the continuous discontinuous Galerkin method. We prove in an abstract setting that standard (continuous) Galerkin finite element approximations are the limit of interior penalty discontinuous Galerkin approximations as the penalty parameter tends to infinity. We then show that by varying the penalization parameter on only a subset of the domain we reach the continuous discontinuous method in the limit. We present numerical experiments illustrating this approach both for equations of non-negative characteristic form (closely related to advection diffusion reaction equations) and to the problem of incompressible miscible displacement. We show that we may practically determine appropriate discontinuous and continuous regions, resulting in a significant reduction of the number of degrees of freedom required to approximate a solution, by using the properties of the discontinuous Galerkin approximation to the advection diffusion reaction equation.
We finally present novel code for implementing the continuous discontinuous Galerkin method in C++
Finite element approximation of high-dimensional transport-dominated diffusion problems
High-dimensional partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the computational challenge of beating the exponential growth of complexity as a function of dimension is exacerbated by the fact that the problem may be transport-dominated. We develop the analysis of stabilised sparse finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate partial differential equations.\ud
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(Presented as an invited lecture under the title "Computational multiscale modelling: Fokker-Planck equations and their numerical analysis" at the Foundations of Computational Mathematics conference in Santander, Spain, 30 June - 9 July, 2005.
Error analysis of the Galerkin FEM in L 2 -based norms for problems with layers: On the importance, conception and realization of balancing
In the present thesis it is shown that the most natural choice for a norm for the analysis of the Galerkin FEM, namely the energy norm, fails to capture the boundary layer functions arising in certain reaction-diffusion problems. In view of a formal Definition such reaction-diffusion problems are not singularly perturbed with respect to the energy norm. This observation raises two questions:
1. Does the Galerkin finite element method on standard meshes yield satisfactory approximations for the reaction-diffusion problem with respect to the energy norm?
2. Is it possible to strengthen the energy norm in such a way that the boundary layers are captured and that it can be reconciled with a robust finite element method, i.e.~robust with respect to this strong norm?
In Chapter 2 we answer the first question. We show that the Galerkin finite element approximation converges uniformly in the energy norm to the solution of the reaction-diffusion problem on standard shape regular meshes. These results are completely new in two dimensions and are confirmed by numerical experiments. We also study certain convection-diffusion problems with characterisitc layers in which some layers are not well represented in the energy norm.
These theoretical findings, validated by numerical experiments, have interesting implications for adaptive methods. Moreover, they lead to a re-evaluation of other results and methods in the literature.
In 2011 Lin and Stynes were the first to devise a method for a reaction-diffusion problem posed in the unit square allowing for uniform a priori error estimates in an adequate so-called balanced norm. Thus, the aforementioned second question is answered in the affirmative. Obtaining a non-standard weak formulation by testing also with derivatives of the test function is the key idea which is related to the H^1-Galerkin methods developed in the early 70s. Unfortunately, this direct approach requires excessive smoothness of the finite element space considered. Lin and Stynes circumvent this problem by rewriting their problem into a first order system and applying a mixed method. Now the norm captures the layers. Therefore, they need to be resolved by some layer-adapted mesh. Lin and Stynes obtain optimal error estimates with respect to the balanced norm on Shishkin meshes. However, their method is unable to preserve the symmetry of the problem and they rely on the Raviart-Thomas element for H^div-conformity.
In Chapter 4 of the thesis a new continuous interior penalty (CIP) method is present, embracing the approach of Lin and Stynes in the context of a broken Sobolev space. The resulting method induces a balanced norm in which uniform error estimates are proven. In contrast to the mixed method the CIP method uses standard Q_2-elements on the Shishkin meshes. Both methods feature improved stability properties in comparison with the Galerkin FEM. Nevertheless, the latter also yields approximations which can be shown to converge to the true solution in a balanced norm uniformly with respect to diffusion parameter. Again, numerical experiments are conducted that agree with the theoretical findings.
In every finite element analysis the approximation error comes into play, eventually. If one seeks to prove any of the results mentioned on an anisotropic family of Shishkin meshes, one will need to take advantage of the different element sizes close to the boundary. While these are ideally suited to reflect the solution behavior, the error analysis is more involved and depends on anisotropic interpolation error estimates.
In Chapter 3 the beautiful theory of Apel and Dobrowolski is extended in order to obtain anisotropic interpolation error estimates for macro-element interpolation. This also sheds light on fundamental construction principles for such operators. The thesis introduces a non-standard finite element space that consists of biquadratic C^1-finite elements on macro-elements over tensor product grids, which can be viewed as a rectangular version of the C^1-Powell-Sabin element. As an application of the general theory developed, several interpolation operators mapping into this FE space are analyzed. The insight gained can also be used to prove anisotropic error estimates for the interpolation operator induced by the well-known C^1-Bogner-Fox-Schmidt element. A special modification of Scott-Zhang type and a certain anisotropic interpolation operator are also discussed in detail. The results of this chapter are used to approximate the solution to a recation-diffusion-problem on a Shishkin mesh that features highly anisotropic elements. The obtained approximation features continuous normal derivatives across certain edges of the mesh, enabling the analysis of the aforementioned CIP method.:Notation
1 Introduction
2 Galerkin FEM error estimation in weak norms
2.1 Reaction-diffusion problems
2.2 A convection-diffusion problem with weak characteristic layers and a Neumann outflow condition
2.3 A mesh that resolves only part of the exponential layer and neglects the weaker characteristic layers
2.3.1 Weakly imposed characteristic boundary conditions
2.4 Numerical experiments
2.4.1 A reaction-diffusion problem with boundary layers
2.4.2 A reaction-diffusion problem with an interior layer
2.4.3 A convection-diffusion problem with characteristic layers and a Neumann outflow condition
2.4.4 A mesh that resolves only part of the exponential layer and neglects the weaker characteristic layers
3 Macro-interpolation on tensor product meshes
3.1 Introduction
3.2 Univariate C1-P2 macro-element interpolation
3.3 C1-Q2 macro-element interpolation on tensor product meshes
3.4 A theory on anisotropic macro-element interpolation
3.5 C1 macro-interpolation on anisotropic tensor product meshes
3.5.1 A reduced macro-element interpolation operator
3.5.2 The full C1-Q2 interpolation operator
3.5.3 A C1-Q2 macro-element quasi-interpolation operator of Scott-Zhang type on tensor product meshes
3.5.4 Summary: anisotropic C1 (quasi-)interpolation error estimates
3.6 An anisotropic macro-element of tensor product type
3.7 Application of macro-element interpolation on a tensor product Shishkin mesh
4 Balanced norm results for reaction-diffusion
4.1 The balanced finite element method of Lin and Stynes
4.2 A C0 interior penalty method
4.3 Galerkin finite element method
4.3.1 L2-norm error bounds and supercloseness
4.3.2 Maximum-norm error bounds
4.4 Numerical verification
4.5 Further developments and summary
Reference
Numerical solution of steady-state groundwater flow and solute transport problems: Discontinuous Galerkin based methods compared to the Streamline Diffusion approach
In this study, we consider the simulation of subsurface flow and solute
transport processes in the stationary limit. In the convection-dominant case,
the numerical solution of the transport problem may exhibit non-physical
diffusion and under- and overshoots. For an interior penalty discontinuous
Galerkin (DG) discretization, we present a -adaptive refinement strategy
and, alternatively, a new efficient approach for reducing numerical under- and
overshoots using a diffusive -projection. Furthermore, we illustrate an
efficient way of solving the linear system arising from the DG discretization.
In -D and -D examples, we compare the DG-based methods to the streamline
diffusion approach with respect to computing time and their ability to resolve
steep fronts
Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems
The central objective of this paper is to develop reduced basis methods for
parameter dependent transport dominated problems that are rigorously proven to
exhibit rate-optimal performance when compared with the Kolmogorov -widths
of the solution sets. The central ingredient is the construction of
computationally feasible "tight" surrogates which in turn are based on deriving
a suitable well-conditioned variational formulation for the parameter dependent
problem. The theoretical results are illustrated by numerical experiments for
convection-diffusion and pure transport equations. In particular, the latter
example sheds some light on the smoothness of the dependence of the solutions
on the parameters
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