+
+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 a posteriori 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.
+
+
+In particular, this work includes following issues
+
+
+ Discretisation:
+
+
Streamline diffusion and discontinuous Galerkin method
+ for a scalar hyperbolic problem: comparison with respect to
+ accuracy and conservation properties
+
Discontinuous Galerkin method for the compressible Euler equations
+
Use of shock-capturing
+
Higher order boundary approximation at reflective boundaries
+
Solution of the nonlinear and the linear problems
+
Mesh generation for airfoil computations
+
+
+
+
+ A posteriori error estimation and adaptivity:
+
+
Derivation (by duality argument) of error estimates with
+ respect to arbitrary target functionals
+
+
Question of well-posedness of the dual problem
+
+
Approximation of the error representation by numerical
+ approximation of the dual solution
+
+
Adaptive algorithm tailored to the efficient computation
+ of the quantity of interest
+
+
+
+
Numerical Examples illustrating the performance of the
+ error estimation and the adaptive grid refinement for a wide range
+ of hyperbolic problems including
+
+
the linear advection equation
+
the 1D inviscid Burgers equation
+
the Buckley-Leverett equation
+
the 1D compressible Euler equations
+
and several examples for the 2D compressible Euler equations.
+
+ and for a variety of target functionals (quantities) including
+
+
drag and lift coefficients of airfoils
+
pressure point values
+
weighted boundary integrals
+
+
+
+
+ Appendices
+
+
Full description of exact solutions to hyperbolic
+ problems treated in numerical examples
+
+
Implementational details of finite elements with curved
+ boundaries
+
+