Research Outputs
An a posteriori error estimate for a dual mixed method applied to Stokes system with non-null source terms
In this work, we focus our attention in the Stokes flow with nonhomogeneous source terms, formulated in dual mixed form. For the sake of completeness, we begin recalling the corresponding well-posedness at continuous and discrete levels. After that, and with the help of a kind of a quasi-Helmholtz decomposition of functions in H (div), we develop a residual type a posteriori error analysis, deducing an estimator that is reliable and locally efficient. Finally, we provide numerical experiments, which confirm our theoretical results on the a posteriori error estimator and illustrate the performance of the corresponding adaptive algorithm, supporting its use in practice.
Numerical analysis of a stabilized scheme applied to incompressible elasticity problems with Dirichlet and with mixed boundary conditions
We analyze a new stabilized dual-mixed method applied to incompressible linear elasticity problems, considering two kinds of data on the boundary of the domain: non homogeneous Dirichlet and mixed boundary conditions. In this approach, we circumvent the standard use of the rotation to impose weakly the symmetry of stress tensor. We prove that the new variational formulation and the corresponding Galerkin scheme are well-posed. We also provide the rate of convergence when each row of the stress is approximated by Raviart-Thomas elements and the displacement is approximated by continuous piecewise polynomials. Moreover, we derive a residual a posteriori error estimator for each situation. The corresponding analysis is quite different, depending on the type of boundary conditions. For known displacement on the whole boundary, we based our analysis on Ritz projection of the error, which requires a suitable quasi-Helmholtz decomposition of functions living in H (div; Ω). As a result, we obtain a simple a posteriori error estimator, which consists of five residual terms, and results to be reliable and locally efficient. On the other hand, when we consider mixed boundary conditions, these tools are not necessary. Then, we are able to develop an a posteriori error analysis, which provides us of an estimator consisting of three residual terms. In addition, we prove that in general this estimator is reliable, and when the traction datum is piecewise polynomial, locally efficient. In the second situation, we propose a numerical procedure to compute the numerical approximation, at a reasonable cost. Finally, we include several numerical experiments that illustrate the performance of the corresponding adaptive algorithm for each problem, and support its use in practice.
A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: The stationary case Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday
This article is concerned with the Stokes system with nonhomogeneous source terms and nonhomogeneous Dirichlet boundary condition. First, we reformulate the problem in its dual mixed form, and then, we study its corresponding well‐posedness. Next, in order to circumvent the well‐known Babuška‐Brezzi condition, we analyze a stabilized formulation of the resulting approach. Additionally, we endow the scheme with an a posteriori error estimator that is reliable and efficient. Finally, we provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.
An open-source system for generating and computer grading traditional non-coding assignments
One of the most time-consuming activities in higher education is reviewing and grading student evaluations. Rapid and effective feedback of evaluations, along with an appropriate assessment strategy, can significantly improve students’ performance. Furthermore, academic dishonesty is a major issue in higher education that has been aggravated by the limitations derived from the COVID-19 pandemic. One of the possible ways to mitigate this issue is to give different evaluations to each student, with the negative cost of increasing reviewing time. In this work, an open-source system developed in Python to automatically create and correct evaluations is presented. Using Jupyter Notebook as the graphical user interface, the system allows the creation of individual student question sheets, with the same structure and different parameter values, to send them to students, grade them, and send the final score back to the students. The proposed system requires little programming knowledge for the instructors to use it. The system was applied in Civil Engineering and Geological Engineering programs at the Universidad Católica de la Santísima Concepción, drastically reducing grading time while improving students’ performance.