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Choice of the SUPG parameter for higher order finite elements
Kohutka, Jiří ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
In this work, we deal with the finite element method Streamline Upwind/Petrov-Galerkin (SUPG) and use it to solve boundary value problem for the stationary convection-diffusion equation with dominant convection with Dirichlet boundary condition on the whole boundary of bounded polyhedral computational domain of dimension 1 and 2, respectively. We consider a quadratic Lagrangian finite elements on the line segments and triangles, respectively. The core of the work is a proposition of choice of stabilizing parameter of SUPG method as an elementwise affine function in outflow boundary layer and as an elementwise constant function in the rest of the computational domain. We show that this choice gives a more accurate solution than the choice of the stabilization parameter as a constant in each element. 1
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor) ; Franz, Sebastian (referee) ; Vejchodský, Tomáš (referee)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Choice of the SUPG parameter for higher order finite elements
Kohutka, Jiří ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
In this work, we deal with the finite element method Streamline Upwind/Petrov-Galerkin (SUPG) and use it to solve boundary value problem for the stationary convection-diffusion equation with dominant convection with Dirichlet boundary condition on the whole boundary of bounded polyhedral computational domain of dimension 1 and 2, respectively. We consider a quadratic Lagrangian finite elements on the line segments and triangles, respectively. The core of the work is a proposition of choice of stabilizing parameter of SUPG method as an elementwise affine function in outflow boundary layer and as an elementwise constant function in the rest of the computational domain. We show that this choice gives a more accurate solution than the choice of the stabilization parameter as a constant in each element. 1
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