Abstract
This thesis summarizes certain boundary element methods
applied to some initial and boundary value problems.
Our model problem is the two-dimensional homogeneous heat conduction
problem with vanishing initial data. We use the heat potential
representation of the solution. The given boundary conditions,
as well as the choice of the representation formula,
yield various boundary integral equations. For the sake of simplicity,
we use the direct boundary integral approach, where
the unknown boundary density appearing in the boundary integral
equation is a quantity of physical meaning.
We consider two different sets of boundary conditions, the Dirichlet problem,
where the boundary temperature is given and the Neumann problem,
where the heat flux across the boundary is given.
Even a nonlinear Neumann condition satisfying certain monotonicity
and growth conditions is possible. The approach yields
a nonlinear boundary integral equation of the second kind.
In the stationary case, the model problem reduces to a potential
problem with a nonlinear Neumann condition. We use the spaces of smoothest
splines as trial functions. The nonlinearity is approximated by using the
L2-orthogonal projection. The resulting collocation scheme retains
the optimal L2-convergence. Numerical experiments are in
agreement with this result.
This approach generalizes to the time dependent case.
The trial functions are tensor products of piecewise linear
and piecewise constant splines. The proposed projection method
uses interpolation with respect to the space variable and the orthogonal
projection with respect to the time variable. Compared to the
Galerkin method, this approach simplifies the realization of the
discrete matrix equations.
In addition, the rate of the convergence is of optimal order.
On the other hand,
the Dirichlet problem, where the boundary temperature is given,
leads to a single layer heat operator equation of the first kind.
In the first approach, we use tensor products of piecewise linear splines
as trial functions with collocation at the nodal points.
Stability and suboptimal L2-convergence of the method were proved in the
case of a circular domain. Numerical experiments indicate the
expected quadratic L2-convergence.
Later, a Petrov-Galerkin approach was proposed, where the trial functions were
tensor products of piecewise linear and piecewise constant splines.
The resulting approximative scheme is stable and
convergent. The analysis has been carried out in the cases of
the single layer heat operator and the hypersingular heat operator.
The rate of the convergence with respect to the L2-norm
is also here of suboptimal order.
| Identifer | oai:union.ndltd.org:oulo.fi/oai:oulu.fi:isbn951-42-5614-X |
| Date | 12 April 2000 |
| Creators | Hamina, M. (Martti) |
| Publisher | University of Oulu |
| Source Sets | University of Oulu |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/doctoralThesis, info:eu-repo/semantics/publishedVersion |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess, © University of Oulu, 2000 |
| Relation | info:eu-repo/semantics/altIdentifier/pissn/0355-3191, info:eu-repo/semantics/altIdentifier/eissn/1796-220X |
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