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+
+<H3>Abstract:</H3>
+<DIR>
+In this thesis we introduce a discontinuous Galerkin method for the
+numerical solution of hyperbolic conversation laws, as for example the
+compressible Euler equations of gas dynamics. Based on this finite
+element method, we develop an adaptive algorithm for the efficient
+computation of physically relevant quantities of the solution. This
+includes <em>a posteriori</em> error estimation of the error in the
+computed quantity as well as adaptive mesh design specifically
+tailored to the efficient computation of this quantity. We illustrate
+this approach by several different hyperbolic problems in combination
+with various different target quantities, including the efficient
+computation of drag and lift coefficients of airfoils immersed in
+inviscid compressible gas flows.
+<br>
+<br>
+In particular, this work includes following issues
+<ul>
+ <li> <p>
+ Discretisation:
+ <ul>
+ <li> Streamline diffusion and discontinuous Galerkin method
+ for a scalar hyperbolic problem: <br> comparison with respect to
+ accuracy and conservation properties
+ <li> Discontinuous Galerkin method for the compressible Euler equations
+ <li> Use of shock-capturing
+ <li> Higher order boundary approximation at reflective boundaries
+ <li> Solution of the nonlinear and the linear problems
+ <li> Mesh generation for airfoil computations
+ </ul>
+ </p>
+
+ <li> <p>
+ A posteriori error estimation and adaptivity:
+ <ul>
+ <li> Derivation (by duality argument) of error estimates with
+ respect to arbitrary target functionals
+
+ <li> Question of well-posedness of the dual problem
+
+ <li> Approximation of the error representation by numerical
+ approximation of the dual solution
+
+ <li> Adaptive algorithm tailored to the efficient computation
+ of the quantity of interest
+ </ul>
+ </p>
+
+ <li> <p> Numerical Examples illustrating the performance of the
+ error estimation and the adaptive grid refinement for a wide range
+ of hyperbolic problems including
+ <ul>
+ <li> the linear advection equation
+ <li> the 1D inviscid Burgers equation
+ <li> the Buckley-Leverett equation
+ <li> the 1D compressible Euler equations
+ <li> and several examples for the 2D compressible Euler equations.
+ </ul>
+ and for a variety of target functionals (quantities) including
+ <ul>
+ <li> drag and lift coefficients of airfoils
+ <li> pressure point values
+ <li> weighted boundary integrals
+ </ul>
+ </p>
+
+ <li> <p>
+ Appendices
+ <ul>
+ <li> Full description of exact solutions to hyperbolic
+ problems treated in numerical examples
+
+ <li> Implementational details of finite elements with curved
+ boundaries
+
+ </ul>
+</DIR>
+<P>
+<BR><HR>
+<ADDRESS>
+<I>Ralf Hartmann</I>
+<BR><I>2002-07-30</I>
+</ADDRESS>
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+
+
+