Abstract: Goal-oriented adaptive algorithms have been widely employed during the last three decades to produce optimal grids in order to solve challenging engineering problems. In this work, we extend the error representation using unconventional dual problems for goal-oriented adaptivity in the context of frequency-domain wave-propagation problems to the case of time-domain problems. To do that, we express the entire problem in weak form in order to formulate the adjoint problem and apply the goal-oriented adaptivity. We have also chosen specific spaces of trial and test functions that allow us to express a classical Method of Lines in terms of a Galerkin scheme. Some numerical results are provided in on espatial dimension which show that the upper bounds of the new error representation are sharper than the classical ones and, therefore, this new error representation can be used to design better goal-oriented adaptive processes.

Abstract: The paper presents an extension of the hypergraph grammar model of the hp-adaptive finite element method algorithm with rectangular elements to the case of non-stationary problems. In our approach the finite element mesh is represented by hypergraphs, the mesh transformations are modelled by means of hypergraph grammar rules. The extension concerns the construction of the elimination tree during the generation of the mesh and mesh adaptation process. Each operation on the mesh (generation of the mesh as well as h-adaptation of the mesh) is followed by the corresponding operation on the elimination tree. The constructed elimination tree allows the solver for reutilization of the matrices computed in the previous step of Finite Element Method. Based on the constructed elimination tree the solver can efficiently solve non-stationary problems.

Abstract: The magnetotelluric (MT) method is a passive electromagnetic (EM) exploration technique governed by Maxwell's equations aiming at estimating the resistivity distribution of the subsurface on scales varying from few meters to hundreds of kilometers. Natural EM sources induce electric currents in the Earth, and these currents generate secondary fields. By measuring simultaneously the horizontal components of these fields on the Earth's surface, it is possible to obtain information about the electrical properties of the subsurface. The dimensionality analysis of MT data is a hot and ongoing research topic in the area. In particular, the work of Weaver et al. (2000) has to be highlighted. There, he presented a dimensionality study based on the rotational invariants of the MT tensor. We also emphasize the more recent work of Martí et al. (2009), who implemented a software (based on these invariants) able to describe in a robust way the dimensionality of the problem when real measurements are employed. The dimension of the formation is not clear in some scenarios. When employing traditional inversion techniques, the dimension (the full 2D (or 3D) problem) is usually fixed in forward simulations and inversion. However, a proper study of the dimensionality of the problem may indicate some areas where the problem is fully 2D (or 3D), while others where a 1D (or 2D) consideration of the problem may be sufficient. Following this idea, we propose an initial step towards an algorithm that takes advantage of this scenario via an adaptivity in the spatial variable. Thus, we first consider a full 1D inverse problem, with an exact (and fast) forward solution, and after that, we introduce this 1D inverse problem solution into the 2D inverse problem. Numerical results show savings in the inversion process of the 75% in some scenarios.

Abstract: In this paper we compare three different strategies for dealing with local singularities with two dimensional isogeometric finite element method. The first strategy employs local h refinements with T-spline basis functions. The second strategy is a modification of the first one, also using local h refinements and T-splines, but with some additional refinements intended to localize the support of the T-spline basis functions. The third strategy utilizes C^0 separators and B-splines. We compare the strategies by means of their computational cost from the point of view of multi-frontal direct solvers. We also compare the computational costs of our strategies with classical FEM using second order polynomials and C^0 separators between elements. We analyse the computational costs theoretically and as well compare the number of floating point operations (FLOPs) executed by the multi-frontal direct solver MUMPS. We show that third strategy outperforms both IGA-FEM and classical FEM.