# Análisis Numérico

Organizadores: Gabriel Acosta (gacosta@dm.uba.ar), Pedro Morin (pmorin@fiq.unl.edu.ar), Rodolfo Rodríguez (rodolfo@ing-mat.udec.cl)

• #### Miércoles 15

15:00 - 15:45

#### Transversally-Enriched Pipe Element Method: an efficient approach for computational hemodynamics

Pablo Blanco (Laboratório Nacional de Computação Científica, Brasil) joint with Alonso M. Alvarez, Raúl A. Feijóo

The development of novel numerical and computational strategies for the simulation of blood flow in complex patient-specific vasculatures has gained strong attention in the scientific community aiming both theoretical and applied research. The need for fast, yet accurate methods is at the core of any translational research enterprise. In this talk, we will introduce a middle fidelity numerical approach to tackle the demanding blood-flow simulation in deformable arteries. This strategy can be placed in-between oversimplified (low-fidelity) reduced 1D models and expensive (high-fidelity) full 3D models. The main feature of the method is the discretization of the pipe-like domain in which the fluid flow problem is addressed. This pipe-oriented discretization is exploited by placing a differentiated polynomial approximation for cross-sectional and longitudinal directions. We will show the basic idea of the method, a numerical study of its convergence properties and applications for patient-specific blood flow modeling in large branching arterial geometries.

15:45 - 16:30

#### Virtual Element Spectral Analysis for the Transmission Eigenvalue Problem.

David Mora (Universidad del Bio-Bio, Concepción, Chile)

The aim of this talk is to analyze a C1 Virtual Element Method (VEM) on polygonal meshes for solving a quadratic and non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained. Numerical experiments will be provided to verify the theoretical error estimates.

16:45 - 17:30

#### LSD for multiscale problems

Alexandre Madureira (Laboratório Nacional de Computação Científica, Brasil)

The Localized Spectral Decomposition finite element method is based on a hybrid formulation of elliptic partial differential equations, that is then transformed via a FETI-like space decomposition. Such decomposition make the formulation embarrassingly parallel and efficient, in particular in the presence of multiscale coefficients. It differs from most of the methods out there since it requires solution's minimum regularity. The major novelty is that we base our decomposition in local spectral problems, resulting in methods that are robust with respect to high contrast coefficients.

This is a joint work with Marcus Sarkis, of WPI (Worcester Polytechnic Institute).

17:30 - 18:15

#### A fully discrete adaptive scheme for parabolic equations

Eduardo Garau (Universidad Nacional del Litoral, Santa Fe, Argentina)

We present an adaptive algorithm for solving linear parabolic equations using hierarchical B-splines and the implicit Euler method for the spatial and time discretizations, respectively.

Our development improves upon one from 2018 by Gaspoz and collaborators, where fully discrete adaptive schemes have been analyzed within the framework of classical finite elements. Our approach is based on an a posteriori error estimation that essentially consists of four indicators: a time and a consistency error indicator that dictate the time-step size adaptation, and coarsening and a space error indicator that are used to obtain suitably adapted hierarchical meshes (at different time steps). Even though we use hierarchical B-splines for the space discretization, a straightforward generalization to other methods, such as FEM, is possible. The algorithm is guaranteed to reach the final time within a finite number of operations, and keep the space-time error below a prescribed tolerance. Some numerical tests document the practical performance of the proposed adaptive algorithm.

• #### Jueves 16

15:00 - 15:45

#### Analysis and approximation of fluids under singular forcing

Enrique Otárola (Universidad Técnica Federico Santamaría, Valparaíso, Chile)

We present a well-posedness theory for Newtonian and some non-Newtonian fluids under singular forcing in Lipschitz domains and in convex polytopes. The main idea, that allows us to deal with such forces, is that we study the fluid problems in suitably weighted Sobolev spaces. We develop an a priori approximation theory, which requires the development of the stability of the Stokes projection over weighted spaces. We conclude by presenting existence results for a singular Bousinessq system and, in the case that the forcing is a linear combination of Dirac deltas, we also present an a posteriori error estimator.

15:45 - 16:30

#### Local error estimates for nonlocal problems

Juan Pablo Borthagaray (Universidad de la República, Uruguay)

The integral fractional Laplacian of order $$s \in (0,1)$$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions.

In this talk we shall discuss regularity of solutions on bounded Lipschitz domains. For finite element discretizations, we derive local error estimates in the $$H^s$$-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.

The talk is based on joint work with Dmitriy Leykekhman (University of Connecticut) and Ricardo Nochetto (University of Maryland).

16:45 - 17:30

#### A priori and a posteriori error analysis of a momentum conservative mixed-FEM for the stationary Navier-Stokes problem

Jessika Camaño (Universidad Católica de la Santísima Concepción, Concepción, Chile)

In this talk we present a new conforming mixed finite element method for the Navier-Stokes problem posed on non-standard Banach spaces, where a pseudostress tensor and the velocity are the main unknowns of the system. The associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the pseudostress and discontinuous piecewise polynomials of degree k for the velocity. Next, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed scheme.

Joint work with S. Caucao (Universidad Católica de la Santísima Concepción, Chile), C. Garcia (Pontificia Universidad Católica de Chile), R. Oyarzúa (Universidad del Bío Bío, Chile) and S. Villa (Universidad del Bío Bío, Chile).

17:30 - 18:15