Linear poroelasticity models have important applications in biology and geophysics. In particular, the well-known Biot consolidation model describes the coupled interaction between the linear response of a porous elastic medium saturated with fluid and a diffusive fluid flow within it, assuming small deformations. This is the starting point for modeling human organs in computational medicine and for modeling the mechanics of permeable
rock in geophysics. Finite element methods for Biot’s consolidation model have been widely studied over the past four decades.
In the first part of the talk, we discuss a posteriori error estimators for locking-free mixed finite element approximation of Biot’s consolidation model. The simplest of these is a conventional residual-based estimator. We establish bounds relating the estimated and true errors, and show that these are independent of the physical parameters. The other two estimators require the solution of local problems. These local problem estimators are also shown to be reliable, efficient and robust. Numerical results are presented that
validate the theoretical estimates, and illustrate the effectiveness of the estimators in guiding adaptive solution algorithms.
In the second part of talk, we discuss a novel locking-free stochastic Galerkin mixed finite element method for the Biot consolidation model with uncertain Young’s modulus and hydraulic conductivity field. After introducing a five-field mixed variational formulation of the standard Biot consolidation model, we discuss stochastic Galerkin mixed finite element approximation, focusing on the issue of well-posedness and efficient linear algebra for the discretized system. We introduce a new preconditioner for use with MINRES and
establish eigenvalue bounds. Finally, we present specific numerical examples to illustrate the efficiency of our numerical solution approach.

Finally, we discuss some remarks related to non-conforming approximation of Biot’s consolidation model.

References:
1. A. Khan, D. J. Silvester, Robust a posteriori error estimation for mixed finite
element approximation of linear poroelsticity, IMA Journal of Numerical Analysis, Oxford University Press, 41 (3), 2021, 2000-2025.
2. A. Khan, C. E. Powell, Parameter-robust stochastic Galerkin approxination for linear poroelasticity with uncertain inputs, SIAM Journal on Scientific Computing (SISC), 43 (4), 2021, B855-B883.
3. A. Khan, P. Zanotti, A nonsymmetric approach and a quasi-optimal and robust discretization for the Biot’s model. Mathematics of Computation, 91 (335), 2022, 1143-1170.
4. V. Anaya, A. Khan, D. Mora, R. Ruiz-Baier, Robust a posteriori error analysis for rotation-based formulations of the elasticity/poroelasticity coupling, SIAM Journal
on Scientific Computing (SISC), 2022.

Understanding the stability of ecological communities is a matter of increasing importance in the context of global environmental change. Yet it has proved to be a challenging task. Different metrics are used to assess the stability of ecological systems, and the choice of one metric over another may result in conflicting conclusions. While the need to consider this multitude of stability metrics has been clearly stated in the ecological literature for decades, little is known about how different stability metrics relate to each other. I’ll present results of dynamical simulations of ecological communities investigating the correlations between frequently used stability metrics, and I will discuss how these results may contribute to make progress in the quantification of stability in theory and in practice.

Zoom Link: https://zoom.us/j/93174968155?pwd=TUJ3WVl1UGNMV0FxQTJQMFY0cjJNdz09

Meeting ID: 931 7496 8155

Passcode: 502784

We build a model of a financial market where a large number of firms determine their dynamic emission strategies under climate transition risk in the presence of both environmentally concerned and neutral investors. The firms aim to achieve a trade-off between financial and environmental performance, while interacting through the stochastic discount factor, determined in equilibrium by the investors' allocations. We formalize the problem in the setting of mean-field games and prove the existence and uniqueness of a Nash equilibrium for firms. We then present a convergent numerical algorithm for computing this equilibrium and illustrate the impact of climate transition risk and the presence of environmentally concerned investors on the market decarbonization dynamics and share prices. We show that uncertainty about future climate risks and policies leads to higher overall emissions and higher spreads between share prices of green and brown companies. This effect is partially reversed in the presence of environmentally concerned investors, whose impact on the cost of capital spurs companies to reduce emissions. However, if future climate policies are uncertain, even a large fraction of environmentally concerned investors is unable to bring down the emission curve: clear and predictable climate policies are an essential ingredient to allow green investors to decarbonize the economy.

Joint work with Pierre Lavigne