Time-Dependent Wave-Structure Interaction Revisited: Thermo-piezoelectric Scatterers

In this paper, we are concerned with a time-dependent transmission problem for a thermo-piezoelectric elastic body immersed in a compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided an appropriate scaling factor is employed. As usual, based on estimates for solutions in the Laplace-transformed domain, we may obtain properties of corresponding solutions in the time-domain without having to perform the inversion of the Laplace-domain solutions

A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems

We present a priori and a posteriori error analysis of a high order hybridizable discontinuous Galerkin (HDG) method applied to a semi-linear elliptic problem posed on a piecewise curved, non polygonal domain. We approximate $\Omega$ by a polygonal subdomain $\Omega_h$ and propose an HDG discretization, which is shown to be optimal under mild assumptions related to the non-linear source term and the distance between the boundaries of the polygonal subdomain $\Omega_h$ and the true domain $\Omega$. Moreover, a local non-linear post-processing of the scalar unknown is proposed and shown to provide an additional order of convergence. A reliable and locally efficient a posteriori error estimator that takes into account the error in the approximation of the boundary data of $\Omega_h$ is also provided

El método de Galerkin discontinuo hibridizable: una aplicación al equilibrio magnético en reactores de fusión.

En reactores de fusión con simetría axial, la condición de equilibrio entre la presión hidrostática en el plasma y la fuerza de confinamiento magnética puede expresarse en términos de la solución de una ecuación semi lineal en derivadas parciales conocida como la ecuación de Grad-Shafranov. La solución de este problema de manera eficiente, rápida y con un alto grado de precisión resulta importante para el diseño de reactores y monitoreo en tiempo real de plasmas en dispositivos experimentales. El método de Galerkin Discontinuo Hibridizable es una estrategia de solución numérica que, basada en una formulación débil de la ecuación, convierte el problema en un conjunto de problemas locales. Estos subproblemas pueden ser resueltos en paralelo y la solución global se construye "pegando" las soluciones locales. Esta estrategia puede tener un orden de aproximación alto y es muy robusta con respecto a las propiedades geométricas del dominio. En esta plática introduciré las ideas básicas del método aplicándolas a la ecuación de Grad-Shafranov.Non UBCUnreviewedAuthor affiliation: New York UniversityPostdoctora

Error Analysis of an Unfitted HDG Method for a Class of Non-linear Elliptic Problems

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.ConiCyT12 month embargo; published: 04 February 2022This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Time-Dependent Wave-Structure Interaction Revisited: Thermo-Piezoelectric Scatterers

In this paper, we are concerned with a time-dependent transmission problem for a thermo-piezoelectric elastic body that is immersed in a compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided that an appropriate scaling factor is employed. As usual, based on estimates for solutions in the Laplace-transformed domain, we may obtain properties of corresponding solutions in the time-domain without having to perform the inversion of the Laplace-domain solutions

Afternote to Coupling at a distance: convergence analysis and a priori error estimates

In their article "Coupling at a distance HDG and BEM", Cockburn, Sayas and Solano proposed an iterative coupling of the hybridizable discontinuous Galerkin method (HDG) and the boundary element method (BEM) to solve an exterior Dirichlet problem. The novelty of the numerical scheme consisted of using a computational domain for the HDG discretization whose boundary did not coincide with the coupling interface. In their article, the authors provided extensive numerical evidence for convergence, but the proof of convergence and the error analysis remained elusive at that time. In this article we fill the gap by proving the convergence of a relaxation of the algorithm and providing a priori error estimates for the numerical solution.Comment: Dedicated to the memory of Francisco--Javier Saya