131 research outputs found
High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates
We present a new AFEM for the Laplace-Beltrami operator with arbitrary
polynomial degree on parametric surfaces, which are globally and
piecewise in a suitable Besov class embedded in with . The idea is to have the surface sufficiently well resolved in
relative to the current resolution of the PDE in . This gives
rise to a conditional contraction property of the PDE module. We present a
suitable approximation class and discuss its relation to Besov regularity of
the surface, solution, and forcing. We prove optimal convergence rates for AFEM
which are dictated by the worst decay rate of the surface error in
and PDE error in .Comment: 51 pages, the published version contains an additional glossar
A Posteriori Error Estimates for Surface Finite Element Methods
Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases.
In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a geometric error that comes from replacing the original surface by a computational mesh. A posteriori error estimates on flat domains only have a Galerkin component. We use residual type error estimators to measure the Galerkin error. The geometric component of our error estimate becomes zero if we consider flat domains, but otherwise has the same order as the residual one. This is different from the available energy norm based error estimates on surfaces, where the importance of the geometric components diminishes asymptotically as the mesh is refined. We use our results to implement an Adaptive Surface Finite Element Method.
An important tool for proving a posteriori error bounds for non smooth functions is the Scott-Zhang interpolant. A refined version of a standard Scott-Zhang interpolation bound is also proved during our analysis. This local version only requires the interpolated function to be in a Sobolev space defined over an element T instead of an element patch containing T.
In the last section we extend our elliptic results to get estimates for the surface heat equation ut − △Γ u = f using the elliptic reconstruction technique
Trace Finite Element Methods for PDEs on Surfaces
In this paper we consider a class of unfitted finite element methods for
discretization of partial differential equations on surfaces. In this class of
methods known as the Trace Finite Element Method (TraceFEM), restrictions or
traces of background surface-independent finite element functions are used to
approximate the solution of a PDE on a surface. We treat equations on steady
and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in
detail. We review the error analysis and algebraic properties of the method.
The paper navigates through the known variants of the TraceFEM and the
literature on the subject
Convergence of Adaptive Finite Element Methods
We develop adaptive finite element methods (AFEMs) for elliptic
problems, and prove their convergence, based on ideas introduced
by D\"{o}rfler \cite{Dw96}, and Morin, Nochetto, and Siebert
\cite{MNS00, MNS02}. We first study an AFEM for general second
order linear elliptic PDEs, thereby extending the results of Morin
et al \cite{MNS00,MNS02} that are valid for the Laplace operator.
The proof of convergence relies on quasi-orthogonality, which
accounts for the bilinear form not being a scalar product,
together with novel error and oscillation reduction estimates,
which now do not decouple. We show that AFEM is a contraction for
the sum of energy error plus oscillation. Numerical experiments,
including oscillatory coefficients and {both coercive and
non-coercive} convection-diffusion PDEs, illustrate the theory and
yield optimal meshes. The role of oscillation control is now more
crucial than in \cite{MNS00,MNS02} and is discussed and documented
in the experiments.
We next introduce an AFEM for the Laplace-Beltrami operator on
graphs in . We first derive a posteriori error
estimates that account for both the energy error in and the
geometric error in due to approximation of the
surface by a polyhedral one. We devise a marking strategy to
reduce the energy and geometric errors as well as the geometric
oscillation. We prove that AFEM is a contraction on a suitably
scaled sum of these three quantities as soon as the geometric
oscillation has been reduced beyond a threshold. The resulting
AFEM converges without knowing such threshold or any constants,
and starting from any coarse initial triangulation. Several
numerical experiments illustrate the theory.
Finally, we introduce and analyze an AFEM for the Laplace-Beltrami
operator on parametric surfaces, thereby extending the results for
graphs. Note that, due to the nature of parametric surfaces, the
geometric oscillation is now measured in terms of the differences
of tangential gradients rather than differences of normals as for
graphs. Numerical experiments with closed surfaces are provided to
illustrate the theory
A posteriori error control for stationary coupled bulk-surface equations
We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind are relevant for applications in engineering, chemistry and in biology like e.g. biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretisation in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm with several benchmark examples
Finite Element Approximation of Eigenvalues and Eigenfunctions of the Laplace-Beltrami Operator
The surface finite element method is an important tool for discretizing and solving elliptic partial differential equations on surfaces. Recently the surface finite element method has been used for computing approximate eigenvalues and eigenfunctions of the Laplace-Beltrami operator, but no theoretical analysis exists to offer computational guidance.
In this dissertation we develop approximations of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator using the surface finite element method. We develop a priori estimates for the eigenvalues and eigenfunctions of the Laplace-Beltrami operator. We then use these a priori estimates to develop and analyze an optimal adaptive method for approximating eigenfunctions of the Laplace-Beltrami operator
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