Research Outputs
A graph-based algorithm for the approximation of the spectrum of the curl operator
We analyze a new algorithm for the finite element approximation of a family of eigenvalue problems for the curl operator that includes, in particular, the approximation of the helicity of a bounded domain. It exploits a tree-cotree decomposition of the graph relating the degrees of freedom of the Lagrangian finite elements and those of the first family of Nédélec finite elements to reduce significantly the dimension of the algebraic eigenvalue problem to be solved. The algorithm is well adapted to domains of general topology. Numerical experiments, including a not simply connected domain with a not connected boundary, are presented in order to assess the performance and generality of the method.
Basis for high order divergence-free finite element spaces
A method classically used in the lower polynomial degree for the construction of a finite element basis of the space of divergence-free functions is here extended to any polynomial degree for a bounded domain without topological restrictions. The method uses graphs associated with two differential operators: the gradient and the divergence, and selects the basis using a spanning tree of the first graph. It can be applied for the two main families of degrees of freedom, weights and moments, used to express finite element differential forms.
Assessment of two approximation methods for the inverse problem of electroencephalography
The goal of this paper is to compare two computational models for the inverse problem of electroencephalography: the localization of brain activity from measurements of the electric potential on the surface of the head. The source current is modeled as a dipole whose localization and polarization has to be determined. Two methods are considered for solving the corresponding forward problems: the so called subtraction approach and direct approach. The former is based on subtracting a fundamental solution, which has the same singular character of the actual solution, and solving computationally the resulting non-singular problem. Instead, the latter consists in solving directly the problem with singular data by means of an adaptive process based on an aposteriori error estimator, which allows creating meshes appropriately refined around the singularity. A set of experimental tests for both, the forward and the inverse problems, are reported. The main conclusion of these tests is that the direct approach combined with adaptivity is preferable when the localization of the dipole is close to an interface between brain tissues with different conductivities.