Eduardo Gomes Dutra do Carmo

Feedback Petrov-Galerkin methods for convection-dominated problems

Abstract In this paper the Petrov-Galerkin method introduced in Vol. 68, pp. 83–95 is adaptively applied to convection-dominated problems. To this end a feedback function is created which increases or decreases the control of the gradient of the approximate solution. This leads to a method with good stability properties close to boundary layers and high accuracy where regular solutions do occur.

A new stabilized finite element formulation for scalar convection–diffusion problems: the streamline and approximate upwind/Petrov–Galerkin method

Abstract A new stabilized and accurate finite element formulation for convection-dominated problems is herein developed. The basis of the new formulation is the choice of a new upwind function. The upwind function chosen for the new method provokes its degeneration into the SUPG or CAU methods, depending on the approximate solution’s regularity. The accuracy and stability of the new formulation for the linear and scalar advection–diffusion equation is demonstrated in several numerical examples.

A discontinuous finite element-based domain decomposition method

In this work, we introduce a somewhat unconventional finite element-based nonoverlapping domain decomposition method (the convenient space decomposition method (CSD)) for parallel computers that can be applied to a wide class of boundary value problems and which can only be implemented efficiently, exclusively with primal variables, by means of the use of discontinuous elements. Interesting aspects and properties are pointed out along with proofs for the most relevant results. Numerical experiments for purely diffusion problems are also presented in order to evaluate the effectiveness of the proposed scheme and to confirm the validity of the related theoretical results.

Consistent discontinuous finite elements in elastodynamics

Finite element discontinuities with respect to time have recently been extremely used in elastodynamic problems due to their natural utilization in combination with adaptive methods and their efficiency in discontinuity capturing techniques for non-smooth problems. In this work, we present some theoretical aspects and numerical results concerning the use of spatial discontinuities in a consistent finite element method for the same class of problems. We first review some formulations for the elastostatic problem and prove two Korn-like inequalities which are very useful for the derivation of convergence rates in Sobolev norms. Next, we present formulations for the dynamic case along with comments on their properties and estimates of convergence rates for smooth solutions, followed by numerical investigations of a typically non-smooth problem involving classical and emerging variational formulations. We also show some numerical experiments with finite element spaces enriched by discontinuous functions other than piecewise Lagrangian polynomials.

Discontinuous finite element formulations applied to cracked elastic domains

The solutions of boundary value problems defined on cracked domains are usually non-smooth in the surroundings of the crack. In this work, we formulate the elasticity problem of a body with such geometric characteristic in a number of equivalent variational alternatives and show that we can take advantage of the theory of discontinuous finite elements in order to approximate its solution in an interesting way at little higher programming cost in comparison with the classical Galerkin method. The idea consists in splitting the global domain into a number of regions in which local mesh refinements are undertaken independently, producing irregular meshes with non-matching elements that are suitable to be used in discontinuous finite element methods. This strategy seems to be attractive to be employed in situations that we know in advance where the critical regions of the domain are located as well as in adaptive techniques.

The validity of the superconvergent patch recovery in discontinuous finite element formulations

In this work, we investigate numerically the possibility of joining the superconvergent patch recovery technique and discontinuous finite element formulations so that adaptive methods involving independent local mesh refinement processes and possibly different polynomial degrees in neighbouring elements can be constructed.

Finite element spaces with discontinuity capturing, Part I: Transport problems with boundary layers

Abstract New finite element spaces are proposed for transport problems to accurately approximate the solution inside the boundary layers. To this end, discontinuity capturing functions are introduced with additional degrees of freedom only in the elements that contain the boundary layers. Numerical results show the high accuracy of the approximate solution in the new spaces with usual meshes.

The CSD domain decomposition method applied to convection–diffusion problems

The CSD technique is a domain decomposition method that was originally proposed for tackling diffusion problems. Attractive theoretical and numerical results were obtained therein motivated the investigation of the use of the CSD in more general families of problems. A brief review of discontinuous Galerkin formulations is brought up here followed by the presentation of the CSD method for convection-diffusion problems. Theoretical aspects are considered and numerical experiments are shown, which evaluate the performance of this technique for the proposed class of problems.