Mixed finite element methods for elasticity have several advantages including being robust in the nearly incompressible limit. Since the 2002 paper by Arnold and Winther there have been several inf-sup stable elasticity finite elements (a space for the stress and displacement) developed in two and three dimensions. However, finding an entire elasticity complex remained challenging in three dimensions. In the last few years, a few discrete elasticity complexes have been constructed. I will discuss our construction that uses macro-triangulations. This is joint work with Snorre Christiansen, Kaibo Hu, Sining Gong, Jay Gopalakrishnan, Michael Neilan.

In recent years, the main challenges in subsurface energy systems (e.g., enhanced geothermal systems and CO2 sequestration) have included issues arising from the multi-physical and multi-scale nature of the problem, as well as uncertainty quantification. Multi-physics involves coupling solid mechanics, fluid mechanics, thermal energy, and chemical reactions, while multi-scale considerations involve relating pore-scale problems to field-scale problems. These problems are complex and require interdisciplinary efforts to achieve meaningful outcomes in research. In this talk, we will focus on the challenges of the multi-physical formulations for solving subsurface applications, and discuss new enriched Galerkin (EG) finite element methods for coupled flow and transport and poro-elasticity systems. The primary goal of the study is to develop computationally efficient and robust numerical methods that are free from oscillations due to a lack of local conservation, maximum principle violations, int-sup issues, and locking effects. The locally conservative enriched Galerkin (LC-EG) method, which will be used to solve the flow problem, is constructed by adding a constant function to each element based on the classical continuous Galerkin methods (CG). The locking-free enriched Galerkin (LF-EG) method adds a piecewise linear vector to the displacement space. These EG methods incorporate well-known discontinuous Galerkin (DG) techniques but use approximation spaces with fewer degrees of freedom than typical DG methods, offering an efficient alternative. We will present a priori error estimates of optimal order and demonstrate, through numerical examples, that the new method is free from oscillations and locking.

Hydrodynamical evolution in a gravitational field arises in many astrophysical and atmospheric problems. In this presentation, we will talk about high order structure preserving methods for the Euler equations under gravitational fields, which can exactly preserve some fundamental continuum properties of the underlying problems in the discrete level. We consider the Euler–Poisson equations in spherical symmetry with an equilibrium state governed by the Lane–Emden equation, and design well-balanced (WB) and total-energy-conserving (TEC) discontinuous Galerkin finite element methods. High order semi-implicit well-balanced asymptotic preserving (AP) finite difference scheme, for all Mach Euler equations with gravitation, may also be discussed. Extensive numerical examples — including a toy model of stellar core-collapse with a phenomenological equation of state that results in core-bounce and shock formation — are provided to verify the well-balanced property, positivity-preserving property, high-order accuracy, total energy conservation and good resolution for both smooth and discontinuous solutions.