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The cell method (CM) or discrete geometric approach (DGA) in the time domain, already introduced by Codecasa et al. in 2008 for the coupled Ampere-Maxwell and Faraday equations, is here recast as a Galerkin Method similar to the finite-element method (FEM). In particular, it is shown to be a mixed method comprising an explicit scheme and two discontinuous Galerkin (DG) FEM spaces formulated on dual meshes, in which each of the two function spaces provides a continuous numerical flux choice for its dual mesh counterpart. The implemented version is shown to compare favorably in terms of accuracy and efficiency with respect to the classic conforming FEM scheme using Whitney elements. When tested on the same tetrahedral mesh, the Courant-Friedrichs-Lewy (CFL) condition for the proposed approach is a factor of 2 less restrictive on the time step with respect to the curl-conforming FEM scheme.

High quality of geometry representation is regarded essential for high-order methods to maintain their high-order accuracy. An agglomerated high-order mesh generating method is investigated in combination with discontinuous Galerkin (DG) method for solving the 3D compressible Euler and Navier-Stokes equations. In this method, a fine linear mesh is first generated by standard commercial mesh generation tools. By taking advantage of an agglomeration method, a quadratic high-order mesh is quickly obtained, which is coarse but provides a high-quality geometry representation, thus very suitable for high-order computations. High-order discretizations are performed on the obtained grids with DG method and the discretized system is treated fully implicitly to obtain steady state solutions. Numerical experiments on several flow problems indicate that the agglomerated high-order mesh works well with DG method in dealing with flow problems of curved geometries. It is also found that with a fully implicit discretized system and a p-sequencing method, the DG method can achieve convergence state within several time steps which shows significant efficiency improvements compared to its explicit counterparts.

This Thesis proposes a combined formulation of the Discontinuous Galerkin Method (DG) with solenoidal basis functions and the eXtended Finite Element Method (XFEM), in order to solve the incompressible Navier-Stokes equations for unsteady flows around a solid object, providing high orders of accuracy in space and time. This DG-XFEM formulation simplifies the meshing process using structured meshes that also do not need to be updated at every time step if the object moves, reducing the computational cost. In the DG-XFEM formulation a fixed structured mesh is used and its elements are classified in three groups, which receive a different treatment. First, the elements inside the solid object are excluded in the calculations since it is treated as a void. Second, the elements belonging to the fluid are calculated as in the DG solenoidal formulation. Third, for the elements cut by the interface integration is modified using XFEM in order to take into account only the fluid region, considering curved integration cells to accurately compute integrals in high-order elements; straight afterwards it is solved again with the DG solenoidal formulation. In the DG solenoidal formulation incompressible flows are first solved for velocity and only part of the pressure's degrees of freedom (hybrid pressure), reducing the overall size of the system to be solved, while the rest of pressure degrees of freedom (interior pressure) is computed as a postprocessing. A numerical validation of the method is given with the simulation of the classical benchmark test of the flow past a cylinder, showing its good performance in several cases tested. En aquesta Tesina es proposa una formulació combinada de dos mètodes numèrics: ”Discontinuous Galerkin Method” (DG) amb funcions de base solenoïdals i ”eXtended Finite Element Method”(XFEM), per tal de resoldre les equacions de Navier-Stokes per a fluxos no estacionaris al voltant d’un objecte sòlid, proporcionant alts ordres de precisió en l’espai i el temps. Aquesta formulació combinada DG-XFEM simplifica el procés de mallat, utilitzant malles estructurades que a més no necessiten ser actualitzades en cada pas de temps si l’objecte es mou, reduint per tant el cost computacional. En la formulació DG-XFEM s’empra una malla estructurada fixa i els seus elements es classifiquen en tres grups, els quals reben tractaments diferents. Primerament, els elements dins del sòlid són exclosos en els càlculs ja que es tracten com a buits. A continuació, els elements que pertanyen al fluid es calculen igual que en la formulació de DG solenoïdal. Per últim, en els elements tallats per la interfase es modifica la integració fent servir XFEM per tal de tenir en compte només la regió de fluid, considerant cèl·lules d’integració corbes per calcular de forma acurada les integrals en elements d’alt ordre; tot seguit es resol de nou emprant la formulació de DG solenoïdal. En la formulació de DG solenoïdal els fluxos incompressibles es resolen en primer lloc per a les velocitats i només una part dels graus de llibertat de la pressió (pressió híbrida), reduint la grandària global del sistema d’equacions a resoldre, mentre que la resta de graus de llibertat (pressió interior) es calculen com un post-procés. El mètode està validat amb la simulació del clàssic problema de test d’un flux de fluid al voltant d’un cilindre, mostrant el seu bon comportament en vàries simulacions.