Resumen de Monotonicity preserving shock capturing techniques for finite elements
The main object of study of this thesis is the development of artificial diffusion shock capturing techniques for continuous and discontinuous Galerkin (cG and dG) approximations of the convection-diffusion problem. Special emphasis is given to the fulfillment of the Discrete Maximum Principle (DMP).
Two artificial diffusion techniques are proposed for the transport problem in cG. They scale the corresponding artificial viscosity according to the variation of the gradient of the discrete solution between elements and one of them is proven to be monotonicity preserving. Both methods are used in combination with linear stabilization to enhance its performance; in particular a novel symmetric projection stabilization technique based on a local Scott-Zhang projector is proposed. The weighting of such detector in order to preserve the monotonicity properties ¿including entropy stability for 1D¿ of the underlying methods is faced. Both shock capturing techniques are shown to outperform other methods in the literature for different sets of numerical tests.
In the dG case a novel definition of the DMP has been provided. One of the gradient jump shock detectors previously used for cG methods has been adapted to this new paradigm and proved to enjoy the DMP property in the one dimensional case. A possible extension to the multidimensional case is proposed.
A DMP-enjoying multidimensional dG method for the convection-diffusion equation is obtained by means of graph-viscosity techniques. The method perturbs the entries of the problem matrix to enforce some properties that lead to a DMP. Appropriate shock detectors are used to weight the perturbation of the problem matrix and the lumping of the Mass matrix, avoiding an excessive smearing of the final solution.
Finally an hp-adaptive technique is proposed to solve the steady convection-diffusion problem. A novel troubled-cell detector based on the evolution of the gradient of the discrete solution along the refinement process is proposed. This troubled-cell detector is able to detect the shock layers in which linear order is enforced. Moreover the application of the artificial viscosity is restricted to such regions. At the same time, high order polynomials are reached through p-refinement in the smooth regions of the solutions.
The performance of all the methods has been tested by means of various numerical tests and the results obtained are provided and commented in the document.
