(Français) Discontinuous Galerkin finite element methods for the control constrained Dirichlet control problem governed by the diffusion equation.
Sorry, this entry is only available in French.
Discontinuous Galerkin finite element methods for the control constrained Dirichlet control problem governed by the diffusion equation.
Divay Garg: Thursday, 30th January 2025 at 10:30 Abstract: We utilize a unified discontinuous Galerkin approach to approximate the control constrained Dirichlet boundary optimal control problem using finite element method over simplicial triangulation. The continuous optimality system obtained from this method simplifies the control constraints into a simplified Signorini type problem, which is then coupled with boundary value problems for the state and co-state variables. The symmetric property of the discrete bilinear forms is required in order to derive the discrete optimality system. The main focus is to derive residual based a posteriori error estimates in the energy norm, where we address the reliability and efficiency of the proposed a posteriori error estimator. The suitable construction of auxiliary problems, continuous and discrete Lagrange multipliers, and intermediate operators are crucial in developing a posteriori error analysis. We have also established optimal a priori error estimates in the energy norm for all the optimal variables (state, co-state, and control) under the appropriate regularity assumptions. Theoretical findings are confirmed and illustrated through numerical results on both uniform and adaptive meshes.
(Français) Adaptive homotopy continuation for relative permeability models in reservoir simulation
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Adaptive homotopy continuation for relative permeability models in reservoir simulation
Peter Moritz von Schultzendorff: Monday, 13th January 2025 at 10:30 Accurate modeling of physical processes requires an appropriate selection of constitutive laws. In physics-based reservoir simulation, constitutive laws, e.g., relative permeabilities are often chosen to be, mathematically speaking, simple functions, not necessarily adhering to physics. The paradigm of hybrid modeling allows the integration of machine learned (ML) constitutive laws. Trained on lab, field, and fine-scale simulation data, ML models represent the underlying physics with high fidelity.Strong nonlinearities in classic (i.e., non-ML) relative permeability have been identified as one of the main sources for convergence issues of nonlinear solvers in reservoir simulation. This issue grows in severance for ML relative permeability models, as their high fidelity to real-world data compromises the mathematically desirable properties of simpler models.In this work, we employ the homotopy continuation (HC) method to recover nonlinear solver robustness for classic relative permeability models. The HC method improves nonlinear solver robustness by first solving a problem with simpler relative permeabilities and then iteratively traversing a solution curve towards the original, more complex problem. To efficiently trace the solution curve, we leverage a posteriori error estimates to design an adaptive HC algorithm that minimizes the total number of solver iterations.We show the current status of our work, both on the theoretical and implementation side, and give an outlook into the application to ML relative permeabilities.
(Français) Analyse du problème des oscillations parasites des méthodes de Volumes Finis pour des écoulements à faible nombre de Mach en mécanique des fluides.
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Analysis of spurious oscillations problem of Finite Volume Methods for low Mach number flows in fluid mechanics
Ibtissem Lannabi: Thursday, 23rd January 2025 at 11:00 This work is devoted to the numerical simulation of low Mach number flows, modeled by the compressible Euler system. Commonly used solvers for discretizing this model are Godunov-type schemes. These schemes exhibit poor performance at low Mach number in terms of efficiency and accuracy.Indeed, when the Mach number tends to zero, material and acoustic waves propagate on two distinct time scales, making temporal discretization challenging.In particular, an explicit scheme is stable under a CFL condition, which depends on the speed of sound, making this criterion very restrictive.Regarding the accuracy problem observed with quadrangular grids, it arises from the fact that the discrete solution fails to converge to the incompressible solution as the Mach number tends to zero.To overcome this accuracy problem, many fixes have been developed and consist in modifying the numerical diffusion of the original scheme. These corrections improve the accuracy of compressible schemes as the Mach number goes to zero. Unfortunately they introduce other problems, such as the appearance of numerical oscillations (checkerboard modes on a Cartesian grid) in the numerical solution, or the damping of acoustic waves as the Mach number goes to zero. Efficiency is also compromised as these schemes are stable under a more restrictive CFL condition compared to the original scheme. In this talk, we propose to study the phenomenon of oscillations that plagues some of the fixes proposed in the literature. We focus on Roe-type fixes, in particular those that reduce the numerical diffusion on the jump of the normal velocity.The asymptotic analysis of these schemes leads to a discretization of a wave system in which the pressure gradient is centered. To better understand the phenomenon, we focus on the linear wave system.We then show that this fix is not TVD, unlike the Godunov scheme,…
(Français) Iterative solvers and optimal complexity of adaptive finite element methods
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Iterative solvers and optimal complexity of adaptive finite element methods
Ani Miraci: Tuesday, 17th December at 11:00 Finite element methods (FEMs) are often used to discretize second-order elliptic partial differential equations (PDEs). While standard FEMs rely on underlying uniform meshes, adaptive FEMs (AFEMs) drive the local mesh-refinement to capture potential singularities of the (unknown) PDE solution (stemming, e.g., from the data or the domain geometry). Crucially, adaptivity is steered by reliable a posteriori error control, often encoded in the paradigm SOLVE — ESTIMATE — MARK — REFINE. AFEMs allow to obtain optimal rates of convergence with respect to the number of degrees of freedom (an improvement to standard FEMs). However, in terms of computational costs, an adaptive algorithm is inherently cumulative in nature: an initial coarse mesh is used as input and exact finite element solutions need to be computed on consecutively refined meshes before a desired accuracy can be ensured. Thus, in practice, one strives instead to achieve optimal complexity, i.e., optimal rate of convergence with respect to the overall computational cost. The core ingredient needed for optimal complexity consists in the use of appropriate iterative solvers to be integrated as the SOLVE module within the adaptive algorithm. More precisely, one requires:(i) a solver whose each iteration is: (a) of linear complexity and (b) contractive;(ii) a-posteriori-steered solver-stopping criterion which allows to discern and balance discretization and solver error;(iii) nested iteration, i.e., the last computed solver-iterate is used as initial guess in the newly-refined mesh. First, we develop an optimal local multigrid for the context of symmetric linear elliptic second order PDEs and a finite element discretization with a fixed polynomial degree p and a hierarchy of bisection-generated meshes with local size h. The solver contracts the algebraic error hp-robustly and comes with a built-in a posteriori estimator equivalent to the algebraic error.Second, the overall adaptive algorithm is then shown…
(Français) A parameter-free HDG method for linear elasticity with strongly symmetric stress
Sorry, this entry is only available in French.
A parameter-free HDG method for linear elasticity with strongly symmetric stress
Lina Zhao: Thursday, 12th December at 11:00 In this talk, we present a parameter-free hybridizable discontinuous Galerkin (HDG) method of arbitrary polynomial orders for the linear elasticity problem, where the symmetry of stress is strongly imposed. The $H(\tdiv;\Omega)$-conforming space is used for the approximation of the displacement and the standard polynomial space is used for the approximation of the stress. The tangential trace of displacement acts as the Lagrange multiplier. The quasi-optimal approximation (up to data-oscillation term) is established for the $L^2$-error of stress and discrete $H^1$-error of displacement with $\lambda$-independent constants without requiring additional regularity assumption. To guide adaptive mesh refinement, $\lambda$-robust a posteriori error estimator is derived. Several numerical experiments will be reported to demonstrate the performance of the proposed scheme.
