## Palestras do PPGMA

O Programa de Pós-Graduação promove palestras regularmente onde pesquisadores do Departamento de Matemática da UFPR, pesquisadores convidados de outras instituições e alunos do Programa apresentam temas de pesquisa. As apresentações são abertas ao público.

As palestras dos anos anteriores podem ser acessadas no link correspondente.

## Palestras

## 2018

- 08/06, às 16h30 no Anfiteatro A

**Denise de Siqueira da UTFPR - Curitiba**

**Mixed finite element approximations of a singular elliptic problem based on some anisotropic and hp-adaptive curved quarter-point elements**

**Resumo**: Mixed finite element methods are applied to a Poisson problem with singularity at a boundary point. The approximation spaces are based on quarter-point elements, the shape functions inheriting the singular behavior of their quadratic geometric maps. Two mesh scenarios are considered, the first one, eight-noded coarse quadrilateral quarter-point elements, placing two mid-side nodes near the singular vertex, radial singularity is exactly captured along element edges, and their refinements reveal shape regular curved meshes. The other one, collapsed quadrilateral quarter-point elements obtained by reducing one of the quadrilateral element edges to the singular point, the radial singularity is captured inside the coarse macro elements as well. Their uniform refinement generates anisotropic meshes, grading towards the singular point. Results for a typical test problem demonstrate a superior effectiveness of the proposed techniques for convergence acceleration, when confronted with usual affine finite elements, for h, p and hp-adaptive refinements. Specially, collapsed quarter-point elements applied to the singular problem reveal accuracy rates equivalent to standard regular contexts, of smooth solutions discretized on uniform affine meshes. - 03/03 às 16h no Anfiteatro A

**Gabriel Haeser - USP**

**Some results on Nino's Conjecture about second-order optimality conditions**

**Resumo**: We prove an extension of Yuan’s lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of Baccari and Trad (SIAM J Optim 15(2):394-408, 2005), where the classical necessary second-order optimality condition is proved under the assumption that the set of Lagrange multipliers is a bounded line segment. We prove the result under the more general assumption that the Hessian of the Lagrangian, evaluated at the vertices of the Lagrange multiplier set, is a matrix set with at most rank 2. We apply the results to prove the classical second-order optimality condition to problems with quadratic constraints and without constant rank of the Jacobian matrix, which settles a new particular case of the conjecture of Andreani, Martínez and Schuverdt (Optim 56:529-542, 2007). Some other new results about this conjecture will also be presented.

- 09/03/2018 às 14h no Anfiteatro A

**Roger Behling (UFSC - Blumenau)**

**On the linear convergence of the circumcentered-reflection method**

**Resumo**: In order to accelerate the Douglas-Rachford method we recently developed the circumcentered-reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.