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Adaptivity in anisotropic finite element calculationsGrosman, Sergej, January 2006 (has links)
Chemnitz, Techn. Univ., Diss., [2006].
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The inf-sup condition for the Bernardi-Fortin-Raugel element on anisotropic meshesApel, Thomas, Nicaise, Serge 31 August 2006 (has links) (PDF)
On a large class of two-dimensional anisotropic meshes, the inf-sup condition (stability) is proved for the triangular and quadrilateral finite element pairs suggested by Bernardi/Raugel and Fortin. As a consequence the pairs ${\cal P}_2-{\cal P}_0$, ${\cal Q}_2-{\cal P}_0$, and ${\cal Q}_2^\prime-{\cal P}_0$ turn out to be stable independent of the aspect ratio of the elements.
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With a new refinement paradigm towards anisotropic adaptive FEM on triangular meshesSchneider, Rene 15 October 2013 (has links) (PDF)
Adaptive anisotropic refinement of finite element meshes allows to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem.
We present a new approach to adaptive refinement and demonstrate that this allows to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.
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The inf-sup condition for the Bernardi-Fortin-Raugel element on anisotropic meshesApel, Thomas, Nicaise, Serge 31 August 2006 (has links)
On a large class of two-dimensional anisotropic meshes, the inf-sup condition (stability) is proved for the triangular and quadrilateral finite element pairs suggested by Bernardi/Raugel and Fortin. As a consequence the pairs ${\cal P}_2-{\cal P}_0$, ${\cal Q}_2-{\cal P}_0$, and ${\cal Q}_2^\prime-{\cal P}_0$ turn out to be stable independent of the aspect ratio of the elements.
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Finite-Elemente-Mortaring nach einer Methode von J. A. Nitsche für elliptische RandwertaufgabenPönitz, Kornelia 11 September 2006 (has links) (PDF)
Viele technische Prozesse führen auf Randwertprobleme mit partiellen
Differentialgleichungen, die mit Finite-Elemente-Methoden näherungsweise
gelöst werden können. Spezielle Varianten dieser Methoden sind
Finite-Elemente-Mortar-Methoden. Sie erlauben das Arbeiten mit an
Teilgebietsschnitträndern nichtzusammenpassenden Netzen, was für
Probleme mit komplizierten Geometrien, Randschichten, springenden
Koeffizienten sowie für zeitabhängige Probleme von Vorteil sein kann.
Ebenso können unterschiedliche Diskretisierungsmethoden in den einzelnen
Teilgebieten miteinander gekoppelt werden.
In dieser Arbeit wird das Finite-Elemente-Mortaring nach einer Methode
von Nitsche für elliptische Randwertprobleme auf zweidimensionalen
polygonalen Gebieten untersucht. Von besonderem Interesse sind dabei
nichtreguläre Lösungen (u \in H^{1+\delta}(\Omega), \delta>0) mit
Eckensingularitäten für die Poissongleichung sowie die Lamé-Gleichung
mit gemischten Randbedingungen. Weiterhin werden singulär gestörte
Reaktions-Diffusions-Probleme betrachtet, deren Lösungen zusätzlich zu
Eckensingularitäten noch anisotropes Verhalten in Randschichten
aufweisen.
Für jede dieser drei Problemklassen wird das Nitsche-Mortaring
dargelegt. Es werden einige Eigenschaften der Mortar-Diskretisierung
angegeben und a-priori-Fehlerabschätzungen in einer H^1-artigen sowie
der L_2-Norm durchgeführt. Auf lokal verfeinerten Dreiecksnetzen können
auch für Lösungen mit Eckensingularitäten optimale Konvergenzordnungen
nach gewiesen werden. Bei den Lösungen mit anisotropen Verhalten werden
zusätzlich anisotrope Dreiecksnetze verwendet. Es werden auch hier
Konvergenzordnungen wie bei klassischen Finite-Elemente-Methoden ohne
Mortaring erreicht. Numerische Experimente illustrieren die Methode und
die Aussagen zur Konvergenz.
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Control constrained optimal control problems in non-convex three dimensional polyhedral domainsWinkler, Gunter 28 May 2008 (has links) (PDF)
The work selects a specific issue from the numerical analysis of
optimal control problems. We investigate a linear-quadratic optimal
control problem based on a partial differential equation on
3-dimensional non-convex domains. Based on efficient solution methods
for the partial differential equation an algorithm known from control
theory is applied. Now the main objectives are to prove that there is
no degradation in efficiency and to verify the result by numerical
experiments.
We describe a solution method which has second order convergence,
although the intermediate control approximations are piecewise
constant functions. This superconvergence property is gained from a
special projection operator which generates a piecewise constant
approximation that has a supercloseness property, from a sufficiently
graded mesh which compensates the singularities introduced by the
non-convex domain, and from a discretization condition which
eliminates some pathological cases.
Both isotropic and anisotropic discretizations are investigated and
similar superconvergence properties are proven.
A model problem is presented and important results from the regularity
theory of solutions to partial differential equation in non-convex
domains have been collected in the first chapters. Then a collection
of statements from the finite element analysis and corresponding
numerical solution strategies is given. Here we show newly developed
tools regarding error estimates and projections into finite element
spaces. These tools are necessary to achieve the main results. Known
fundamental statements from control theory are applied to the given
model problems and certain conditions on the discretization are
defined. Then we describe the implementation used to solve the model
problems and present all computed results.
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Control constrained optimal control problems in non-convex three dimensional polyhedral domainsWinkler, Gunter 20 March 2008 (has links)
The work selects a specific issue from the numerical analysis of
optimal control problems. We investigate a linear-quadratic optimal
control problem based on a partial differential equation on
3-dimensional non-convex domains. Based on efficient solution methods
for the partial differential equation an algorithm known from control
theory is applied. Now the main objectives are to prove that there is
no degradation in efficiency and to verify the result by numerical
experiments.
We describe a solution method which has second order convergence,
although the intermediate control approximations are piecewise
constant functions. This superconvergence property is gained from a
special projection operator which generates a piecewise constant
approximation that has a supercloseness property, from a sufficiently
graded mesh which compensates the singularities introduced by the
non-convex domain, and from a discretization condition which
eliminates some pathological cases.
Both isotropic and anisotropic discretizations are investigated and
similar superconvergence properties are proven.
A model problem is presented and important results from the regularity
theory of solutions to partial differential equation in non-convex
domains have been collected in the first chapters. Then a collection
of statements from the finite element analysis and corresponding
numerical solution strategies is given. Here we show newly developed
tools regarding error estimates and projections into finite element
spaces. These tools are necessary to achieve the main results. Known
fundamental statements from control theory are applied to the given
model problems and certain conditions on the discretization are
defined. Then we describe the implementation used to solve the model
problems and present all computed results.
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Adaptivity in anisotropic finite element calculationsGrosman, Sergey 09 May 2006 (has links) (PDF)
When the finite element method is used to solve boundary value problems, the
corresponding finite element mesh is appropriate if it is reflects the behavior of the true solution. A posteriori error estimators are suited to construct adequate meshes. They are useful to measure the quality of an approximate solution and to design adaptive solution algorithms. Singularly perturbed problems yield in general solutions with anisotropic features, e.g. strong boundary or interior layers. For such problems it is useful to use anisotropic meshes in order to reach maximal order of convergence. Moreover, the quality of the numerical solution rests on the robustness of the a posteriori error estimation with respect to both the anisotropy of the mesh and the perturbation parameters.
There exist different possibilities to measure the a posteriori error in the energy norm for the singularly perturbed reaction-diffusion equation. One of them is the equilibrated residual method which is known to be robust as long as one solves auxiliary local Neumann problems exactly on each element. We provide a basis for an approximate solution of the aforementioned auxiliary problem and show that this approximation does not affect the quality of the error estimation.
Another approach that we develope for the a posteriori error estimation is the hierarchical error estimator. The robustness proof for this estimator involves some stages including the strengthened Cauchy-Schwarz inequality and the error reduction property for the chosen space enrichment.
In the rest of the work we deal with adaptive algorithms. We provide an overview of the existing methods for the isotropic meshes and then generalize the ideas for the anisotropic case. For the resulting algorithm the error reduction estimates are proven for the Poisson equation and for the singularly perturbed reaction-difussion equation. The convergence for the Poisson equation is also shown.
Numerical experiments for the equilibrated residual method, for the hierarchical
error estimator and for the adaptive algorithm confirm the theory. The adaptive
algorithm shows its potential by creating the anisotropic mesh for the problem
with the boundary layer starting with a very coarse isotropic mesh.
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Adaptivity in anisotropic finite element calculationsGrosman, Sergey 21 April 2006 (has links)
When the finite element method is used to solve boundary value problems, the
corresponding finite element mesh is appropriate if it is reflects the behavior of the true solution. A posteriori error estimators are suited to construct adequate meshes. They are useful to measure the quality of an approximate solution and to design adaptive solution algorithms. Singularly perturbed problems yield in general solutions with anisotropic features, e.g. strong boundary or interior layers. For such problems it is useful to use anisotropic meshes in order to reach maximal order of convergence. Moreover, the quality of the numerical solution rests on the robustness of the a posteriori error estimation with respect to both the anisotropy of the mesh and the perturbation parameters.
There exist different possibilities to measure the a posteriori error in the energy norm for the singularly perturbed reaction-diffusion equation. One of them is the equilibrated residual method which is known to be robust as long as one solves auxiliary local Neumann problems exactly on each element. We provide a basis for an approximate solution of the aforementioned auxiliary problem and show that this approximation does not affect the quality of the error estimation.
Another approach that we develope for the a posteriori error estimation is the hierarchical error estimator. The robustness proof for this estimator involves some stages including the strengthened Cauchy-Schwarz inequality and the error reduction property for the chosen space enrichment.
In the rest of the work we deal with adaptive algorithms. We provide an overview of the existing methods for the isotropic meshes and then generalize the ideas for the anisotropic case. For the resulting algorithm the error reduction estimates are proven for the Poisson equation and for the singularly perturbed reaction-difussion equation. The convergence for the Poisson equation is also shown.
Numerical experiments for the equilibrated residual method, for the hierarchical
error estimator and for the adaptive algorithm confirm the theory. The adaptive
algorithm shows its potential by creating the anisotropic mesh for the problem
with the boundary layer starting with a very coarse isotropic mesh.
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Finite-Elemente-Mortaring nach einer Methode von J. A. Nitsche für elliptische RandwertaufgabenPönitz, Kornelia 29 June 2006 (has links)
Viele technische Prozesse führen auf Randwertprobleme mit partiellen
Differentialgleichungen, die mit Finite-Elemente-Methoden näherungsweise
gelöst werden können. Spezielle Varianten dieser Methoden sind
Finite-Elemente-Mortar-Methoden. Sie erlauben das Arbeiten mit an
Teilgebietsschnitträndern nichtzusammenpassenden Netzen, was für
Probleme mit komplizierten Geometrien, Randschichten, springenden
Koeffizienten sowie für zeitabhängige Probleme von Vorteil sein kann.
Ebenso können unterschiedliche Diskretisierungsmethoden in den einzelnen
Teilgebieten miteinander gekoppelt werden.
In dieser Arbeit wird das Finite-Elemente-Mortaring nach einer Methode
von Nitsche für elliptische Randwertprobleme auf zweidimensionalen
polygonalen Gebieten untersucht. Von besonderem Interesse sind dabei
nichtreguläre Lösungen (u \in H^{1+\delta}(\Omega), \delta>0) mit
Eckensingularitäten für die Poissongleichung sowie die Lamé-Gleichung
mit gemischten Randbedingungen. Weiterhin werden singulär gestörte
Reaktions-Diffusions-Probleme betrachtet, deren Lösungen zusätzlich zu
Eckensingularitäten noch anisotropes Verhalten in Randschichten
aufweisen.
Für jede dieser drei Problemklassen wird das Nitsche-Mortaring
dargelegt. Es werden einige Eigenschaften der Mortar-Diskretisierung
angegeben und a-priori-Fehlerabschätzungen in einer H^1-artigen sowie
der L_2-Norm durchgeführt. Auf lokal verfeinerten Dreiecksnetzen können
auch für Lösungen mit Eckensingularitäten optimale Konvergenzordnungen
nach gewiesen werden. Bei den Lösungen mit anisotropen Verhalten werden
zusätzlich anisotrope Dreiecksnetze verwendet. Es werden auch hier
Konvergenzordnungen wie bei klassischen Finite-Elemente-Methoden ohne
Mortaring erreicht. Numerische Experimente illustrieren die Methode und
die Aussagen zur Konvergenz.
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