Efficient solvers for hybridized three-field mixed finite element coupled poromechanics

Uniformly well-posed hybridized discontinuous Galerkin/hybrid mixed discretizations for Biot's consolidation model

We consider the quasi-static Biot's consolidation model in a three-field formulation with the three unknown physical quantities of interest being the displacement $\boldsymbol{u}$ of the solid matrix, the seepage velocity $\boldsymbol{v}$ of the fluid and the pore pressure $p$. As conservation of fluid mass is a leading physical principle in poromechanics, we preserve this property using an $\boldsymbol{H}(\operatorname{div})$-conforming ansatz for $\boldsymbol{u}$ and $\boldsymbol{v}$ together with an appropriate pressure space. This results in Stokes and Darcy stability and exact, that is, p…

新的混合有限元方法用于线弹性与最优多级求解器

New Hybridized Mixed Methods for Linear Elasticity and Optimal Multilevel Solvers

本文提出新的混合有限元方法用于二维和三维线弹性问题，通过混合化实现应力的符合性和强对称性，采用P_{k+1}-P_k混合格式，保证收敛性，并开发多级求解器验证理论结果。

Stabilized material point methods for coupled large deformation and fluid flow in porous materials

The material point method (MPM) has been increasingly used for the simulation of large deformation processes in fluid-infiltrated porous materials. For undrained poromechanical problems, however, standard MPMs are numerically unstable because they use low-order interpolation functions that violate the inf-sup stability condition. In this work, we develop stabilized MPM formulations for dynamic and quasi-static poromechanics that permit the use of standard low-order interpolation functions notwithstanding the drainage condition. For the stabilization of both dynamic and quasi-static formulation…

一种用于含摩擦接触的基质-裂缝界面混合维孔隙弹性力学的泡状VEM-全离散多面体格式

A bubble VEM-fully discrete polytopal scheme for mixed-dimensional poromechanics with frictional contact at matrix fracture interfaces

本文提出一种结合泡状VEM与混合有限体积的混合维孔隙弹性力学离散格式，用于处理含摩擦接触的裂缝网络，并保证数值稳定性与耗散性。

孔隙力学显式耦合模拟中的压力稳定性及其在CO$_2$封存中的应用

Pressure stability in explicitly coupled simulations of poromechanics with application to CO$_2$ sequestration

研究非迭代固定应力分裂法在近未排水不可压缩孔隙力学问题中的压力稳定效应，发现其需大时间步长且在排水区易致误差，证明压力跳跃稳定在显式与隐式设置中均有效。

A macroelement stabilization for multiphase poromechanics

Strong coupling between geomechanical deformation and multiphase fluid flow appears in a variety of geoscience applications. A common discretization strategy for these problems is a continuous Galerkin finite element scheme for the momentum balance equations and a finite volume scheme for the mass balance equations. When applied within a fully-implicit solution strategy, however, this discretization is not intrinsically stable. In the limit of small time steps or low permeabilities, spurious oscillations in the pressure field, i.e. checkerboarding, may be observed. Further, eigenvalues associa…

连续与间断伽辽金方法与快速多网格求解器的性能比较

A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers

本文比较了连续有限元方法、对称内罚间断伽辽金方法和混合型间断伽辽金方法的性能，研究了高阶方法结合最新多网格求解器在泊松方程中的实现，包括四边形和六面体元素上的快速矩阵自由实现与求和因子化。对于混合型间断伽辽金方法，开发了一种结合从迹空间到线性有限元空间的网格传输与进一步层次上的代数多网格的多网格方法。尽管矩阵型HDG求解器和矩阵自由几何多网格方案在求解器复杂度上相似，但后者在包含超收敛效应后仍能提供数量级更快的求解时间，这是因为矩阵自由操作符评估相对于稀疏矩阵向量乘积有显著更好的性能。屋顶性能模型验证了矩阵自由实现的优势。

Stokes 方程的嵌入式-杂交化间断 Galerkin 有限元方法

An embedded--hybridized discontinuous Galerkin finite element method for the Stokes equations

本文提出一种用于 Stokes 问题的嵌入式-杂交化间断 Galerkin 有限元方法，在保持速度场 H(div)-协调和误差估计与压力无关等优点的同时，减少了全局自由度并改善了线性系统结构，数值实验验证了其收敛性和计算效率优势。

eHDG:An Exponentially Convergent Iterative Solver for HDG Discretizations of Hyperbolic Partial Differential Equations

We present a scalable and efficient iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of hyperbolic partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations. In particular, the method is a fixed-point approach that requires only independent element-by-element local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. We rigorously show that the proposed method is exponentially convergent in the number of iterations for transport and l…

The Hybrid Discontinuous Galerkin method for elliptic problems and applications in vertical ocean-slice modeling

Element Method. The Finite Volume Method guarantees local and global mass conservation. A property not satisfied by the Finite Volume Method. On the down side, the Finite Volume Method requires non trivial modifications to attain high order approximations unlike the Finite Volume Method. It has been contended that the Discontinuous Galerkin Method, locally conservative and high order, is a natural progression for Coastal Ocean Modeling. Consequently, as a primer we consider the vertical ocean-slice model with the inclusion of density effects. To solve these non steady Partial Differential Equa…

High-Order Mixed Finite Element Variable Eddington Factor Methods

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Discrete Ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calcu…