Adaptive Finite Element Methods
Many problems in physics and engineering are formulated in terms of partial differential equations. Since explicit solutions are rarly known, there is great interest in solving these equations numerically.
This is done by discretising the underlying spatial domain (usually in two or three space dimensions) and establishing discrete equations by using the finite difference or the finite element method for example. For timedependent problems one has in addition to perform discrete time steps.
Although speed and storage capabilities of computers have increased in the last decades many practical problems are still only tractable with severe simplifications. Moreover, there is also need to judge the quality of the obtained approximated solution to ensure its reliabilty. So one is interested to use as less as possible degrees of freedom in the problem while on the other hand the approximated solution should be of a certain accuracy.
For this the concept of a posteriori error estimation and adaptive mesh refinement has been developed. This can be done at best within the Finite Element context.
1. The adaptive algorithm (stationary problems)
Assume that we are given some coarse discretisation
(macro discretisation) of the domain
of the problem. In a first step we adjust data to obtain an
initial discretisation.
Now we iterate the procedure
until the a posteriori error is below a prescribed quantity.
Our contributions.
2. Solve
3. A posteriori error estimate
4. Local error indicators
5. Adaptive techniques
6. Grid generation
7. Initial discretisation
8. Remesh
We have to solve a linear or non-linear system of equations. For linear
problems one may use Multigrid or Krylovspace methods, while for
nonlinear problems Newtons method may be used. In the analysis one
often requires that the discrete problem is solved exactly. In
practice, however, one only computes again an approximation and it is
important to match the stopping criterion for this solver with the
discretisation error (measured by the actual error estimation).
Literature:
Our contributions.
The mathematical basis to control the error is an estimation of the
error in terms of available quantities such as the discrete solution,
grid sizes and data. For stationary problems it is of the form
Here u and u_h are the exact and the discrete solution,
'data' and `data_h' stand for data functions
and their discrete realisation. ||.|| and ||.||' are some norms,
||.|| is often the `energy norm' of the problem.
The constant in front of the inequality depends on the
interpolation constant C_i and on the norm of the linearized
operator, the stability constant C_s.
While it is not difficult to provide estimates for C_i,
it is delicate to get information on C_s.
Literature:
Our contributions.
The term E in the inequality (1)
can usually be localised in the following way:
define quantities E_T to each
element T of the discretisation such that
E = | {E_T} |, where | . | is some given vector norm
and {E_T} the vector consisting of all
local error indicators.
For some often used error indicators one can prove the estimate
where S_T is the domain composed of T and all its
neighbours.
Literature:
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Here we give an overview of the different techniques to adapt the
Finite Element space (for stationary problems). These are
Literature:
Our contributions.
The main items are
The first task is in general very complicated. Starting from a
certain description of the domain in question one has to
establish a discretisation (often, a decomposition into
simplices or quadrilaterals) that satisfies some 'regularity'
requirements. For more information see the link below.
Now assume that we already have a discretisation and that we have a
list of marked elements (simplices) that have to be refined.
This means, that any marked simplex should be replaced by simplices
that are of (prescribed) smaller diameter. The problems are to avoid
increasing degeneration of the simplices during repeated refinement
and the 'matching' between the coarse and fine zones.
If the computational domain is not identical to the physical domain
from the continuous problem, we have to further improve the boundary
approximation. In case the discretisation is still coarse, we
refer to the first task again, whereas if the discretisation has
reached a certain fineness, grid refinement is followed by a
relocation of the new nodes at the boundary of the computational
doamin towards the physical boundary.
Literature:
Links:
Our contributions.
In case of non-sufficiently resolved data, the adaptive procedure
may fail. Therefore it is necessary to control this contributions
and since some data terms are of a priori nature it is conveniant
to do it in advance. Thus we apply the concept of local error
indicators to data error, that is we let
for all T and perform the steps estimate
and remesh as in the adaptive algorithm, starting from the
macro discretisation. The evaluation
of E_T has to be performed either by random quadrature
or higher order quadrature rules.
Literature:
Our contributions.
Given a vector of local error indicators {E_T}, we have to
construct a new discretisation on which a smaller error can be
expected. In a first step we construct a list of marked elements, that
is an integer field {m_T} which proposes the number of local
elementary refinement steps (see grid
generation)
that should be performed on T.
From this the new grid has to be constructed.
Marking strategies are:
where \theta is a fixed number in (0,1).
Literature:
Our contributions.
E. Bänsch:
E. Bänsch:
E. Bänsch:
E. Bänsch:
E. Bänsch:
E. Bänsch,
W. Dörfler:
E. Bänsch,
K. Mikula
E. Bänsch,
K.G. Siebert:
W. Dörfler:
W. Dörfler:
W. Dörfler:
W. Dörfler, M. Rumpf:
W. Dörfler, O. Wilderotter:
D. Kröner,
M. Ohlberger:
R.H. Nochetto,
A. Schmidt,
C. Verdi:
R.H. Nochetto,
A. Schmidt,
C. Verdi:
R.H. Nochetto,
A. Schmidt,
C. Verdi:
R.H. Nochetto,
A. Schmidt,
C. Verdi:
M. Ohlberger:
M. Ohlberger:
M. Ohlberger:
A. Schmidt,
K.G. Siebert:
A. Schmidt,
K.G. Siebert:
K.G. Siebert:
K.G. Siebert: