Error estimation for hybrid-H1 finite element approximations and comparaison between hybrid-H1 and H(div) approximations

• Grant number: 19/17186-6
• Support Opportunities: Scholarships in Brazil - Master
• Start date: October 01, 2019
• End date: September 30, 2021
• Field of knowledge: Engineering - Civil Engineering - Structural Engineering
• Agreement: Equinor (former Statoil)
• Principal Investigator: Philippe Remy Bernard Devloo
• Grantee: Victor Bringhenti Oliari
• Host Institution: Faculdade de Engenharia Civil, Arquitetura e Urbanismo (FEC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Company: Universidade Estadual de Campinas (UNICAMP). Faculdade de Engenharia Mecânica (FEM)
• Associated research grant: 17/15736-3 - Engineering Research Centre in Reservoir and Production Management, AP.PCPE

Abstract

The Finite Element Method (FEM) stands out as a numerical method to solve partial differential equations due to its generality, robustness and applicability in various fields. When handling problems with strong gradients, such as simulations of reservoirs with highly heterogeneous porosity, Hybrid FEM formulations excel. In these formulations, the continuity restraint between elements is substituted by a space of Lagrange multipliers. This method also ensures local mass conservation and enables internal degrees of freedom to be statically condensed, reducing the global linear system to be solved. As in every numerical method, the obtained solution is an approximation of the exact unknown solution. An efficient way of reducing the approximation error is locating the elements that contribute the most to the error and applying hp-refinement. [2]. In this context, a posteriori error estimators emerge as a method of locating the elements to be refined, since they evaluate the error at an element level. With that being said, this project's objective is to develop an error estimator for Hybrid FEM approximations based on [1] and [3] and to apply it on fractured reservoir simulations. The estimator is based on the reconstruction of the FEM solution. The reconstructed solution is then used in the Prager-Synge Theorem, which provides an orthogonality relationship that leads to an upper bound for the unknown exact error. (AU)