arXiv subjects

Jeonghun J. Lee

Publications and source records attributed to Jeonghun J. Lee.

36 records · Page 2Linked to original sources

arXiv2022

Analysis of an embedded-hybridizable discontinuous Galerkin method for Biot's consolidation model

We present an embedded-hybridizable discontinuous Galerkin finite element method for the total pressure formulation of the quasi-static poroelasticity model. Although the displacement and the Darcy velocity are approximated by discontinuous piece-wise polynomials, $H(\text{div})$-conformity of these unknowns is enforced by Lagrange multipliers. The semi-discrete problem is shown to be stable and the fully discrete problem is shown to be well-posed. Additionally, space-time a priori error estimates are derived, and confirmed by numerical examples, that show that the proposed discretization is free of volumetric locking.

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arXiv2020

Parameter robust preconditioning by congruence for multiple-network poroelasticity

The mechanical behaviour of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or multi-porosity/multi-permeability systems. These equations generalize Biot's equations, which describe the mechanics of the one-network case. The efficient numerical solution of the MPET equations is challenging, in part due to the complexity of the system and in part due to the presence of interacting parameter regimes. In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically. In particular, we introduce a new approach to formulating finite element methods and associated preconditioners for the MPET equations. The approach is based on designing transformations of variables that simultaneously diagonalize (by congruence) the equations' key operators and subsequently constructing parameter-robust block-diagonal preconditioners for the transformed system. Our methodology is supported by theoretical considerations as well as by numerical results.

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arXiv2020

A mixed method for time-transient acoustic wave propagation in metamaterials

In this paper we develop a finite element method for acoustic wave propagation in Drude-type metamaterials. The governing equation is written as a symmetrizable hyperbolic system with auxiliary variables. The standard mixed finite elements and discontinuous finite elements are used for spatial discretization, and the Crank-Nicolson scheme is used for time discretization. The a priori error analysis of fully discrete scheme is carried out in details. Numerical experiments illustrating the theoretical results and metamaterial wave propagation, are included.

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arXiv2019

Analysis of hybridized discontinuous Galerkin methods without elliptic regularity assumptions

In this paper we present new stability and optimal error analyses of hybridized discontinuous Galerkin (HDG) methods which do not require elliptic regularity assumptions. To obtain error estimates without elliptic regularity assumptions, we use new inf-sup conditions based on stabilized saddle point structures of HDG methods. We show that this approach can be applied to obtain optimal error estimates of HDG methods for the Poisson equations, the convection-reaction-diffusion equations, the Stokes equations, and the Oseen equations.

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arXiv2019

A hybridized discontinuous Galerkin method for Poisson-type problems with sign-changing coefficients

In this paper, we present a hybridized discontinuous Galerkin (HDG) method for Poisson-type problems with sign-changing coefficients. We introduce a sign-changing stabilization parameter that results in a stable HDG method independent of domain geometry and the ratio of the negative and positive coefficients. Since the Poisson-type problem with sign-changing coefficients is not elliptic, standard techniques with a duality argument to analyze the HDG method cannot be applied. Hence, we present a novel error analysis exploiting the stabilized saddle-point problem structure of the HDG method. Numerical experiments in two dimensions and for varying polynomial degree verify our theoretical results.

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arXiv2019

High order approximation of Hodge Laplace problems with local coderivatives on cubical meshes

In mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex, the exterior derivative operators are computed exactly, so the spatial locality is preserved. However, the numerical approximations of the associated coderivatives are nonlocal and it can be regarded as an undesired effect of standard mixed methods. For numerical methods with local coderivatives a perturbation of low order mixed methods in the sense of variational crimes has been developed for simplicial and cubical meshes. In this paper we extend the low order method to all high orders on cubical meshes using a new family of finite element differential forms on cubical meshes. The key theoretical contribution is a generalization of the linear degree, in the construction of the serendipity family of differential forms, and the generalization is essential in the unisolvency proof of the new family of finite element differential forms.

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arXiv2019

A space-time hybridizable discontinuous Galerkin method for linear free-surface waves

We present and analyze a novel space-time hybridizable discontinuous Galerkin (HDG) method for the linear free-surface problem on prismatic space-time meshes. We consider a mixed formulation which immediately allows us to compute the velocity of the fluid. In order to show well-posedness, our space-time HDG formulation makes use of weighted inner products. We perform an a priori error analysis in which the dependence on the time step and spatial mesh size is explicit. We provide two numerical examples: one that verifies our analysis and a wave maker simulation.

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arXiv2019

Unconditionally stable second order convergent partitioned methods for multiple-network poroelasticity

In this paper, we consider partitioned numerical methods for quasi-static multiple-network poroelasticity (MPET) equations, generalizations of the Biot model in poroelasticity for multiple pore networks. Two partitioned numerical methods are presented for the equations which split time discretization into solving two subequations, a Lame equation and a system of heat equations, alternatively. In contrast to the iterative coupling methods which require multiple iterations at each time step, our numerical methods solve these smaller equations only once at each time step. We prove their unconditional stability and high order convergence in time with a novel error analysis. A number of numerical results are presented to illustrate good performances of these partitioned methods.

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arXiv2019

Analysis of an HDG Method for Linearized Incompressible Resistive MHD Equations

We present a hybridized discontinuous Galerkin (HDG) method for stationary linearized incompressible magnetohydrodynamics (MHD) equations. At the heart of the paper is the introduction of an HDG flux of the dual saddle-point form of the MHD equations that facilitates the hybridization of discontinuous Galerkin (DG) method. We carry out the $\textit{a priori}$ error estimates for the proposed HDG method on simplicial meshes in both two- and three-dimensions. The analysis provides optimal convergence for the fluid velocity and the magnetic variables, and quasi-optimal convergence for the remaining quantities. Numerical examples are presented to verify the theoretical findings.

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arXiv2018

Analysis and preconditioning of parameter-robust finite element methods for Biot's consolidation model

In this paper we consider a three-field formulation of the Biot model which has the displacement, the total pressure, and the pore pressure as unknowns. For parameter-robust stability analysis, we first show a priori estimates of the continuous problem with parameter-dependent norms. Then we study finite element discretizations which provide parameter-robust error estimates and preconditioners. For finite element discretizations we consider standard mixed finite element as well as stabilized methods for the Stokes equations, and the complete error analysis of semidiscrete solutions is given. Abstract forms of parameter-robust preconditioners are investigated by the operator preconditioning approach. The theoretical results are illustrated with numerical experiments.

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arXiv2018

A mixed finite element method for nearly incompressible multiple-network poroelasticity

In this paper, we present and analyze a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity (MPET) equations. The MPET equations describe flow and deformation in an elastic porous medium that is permeated by multiple fluid networks of differing characteristics. As such, the MPET equations represent a generalization of Biot's equations, and numerical discretizations of the MPET equations face similar challenges. Here, we focus on the nearly incompressible case for which standard mixed finite element discretizations of the MPET equations perform poorly. Instead, we propose a new mixed finite element formulation based on introducing an additional total pressure variable. By presenting energy estimates for the continuous solutions and a priori error estimates for a family of compatible semi-discretizations, we show that this formulation is robust in the limits of incompressibility, vanishing storage coefficients, and vanishing transfer between networks. These theoretical results are corroborated by numerical experiments. Our primary interest in the MPET equations stems from the use of these equations in modelling interactions between biological fluids and tissues in physiological settings. So, we additionally present physiologically realistic numerical results for blood and tissue fluid flow interactions in the human brain.

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arXiv2017

Local coderivatives and approximation of Hodge Laplace problems

The standard mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex are based on proper discrete subcomplexes. As a consequence, the exterior derivatives, which are local operators, are computed exactly. However, the approximations of the associated coderivatives are nonlocal. In fact, this nonlocal property is an inherent consequence of the mixed formulation of these methods, and can be argued to be an undesired effect of these schemes. As a consequence, it has been argued, at least in special settings, that more local methods may have improved properties. In the present paper, we construct such methods by relying on a careful balance between the choice of finite element spaces, degrees of freedom, and numerical integration rules. Furthermore, we establish key convergence estimates based on a standard approach of variational crimes.

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arXiv2017

Weakly imposed symmetry and robust preconditioners for Biot's consolidation model

We discuss the construction of robust preconditioners for finite element approximations of Biot's consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger-Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.

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arXiv2016

Robust error analysis of coupled mixed methods for Biot's consolidation model

We study the a priori error analysis of finite element methods for Biot's consolidation model. We consider a formulation which has the stress tensor, the fluid flux, the solid displacement, and the pore pressure as unknowns. Two mixed finite elements, one for linear elasticity and the other for mixed Poisson problems are coupled for spatial discretization, and we show that any pair of stable mixed finite elements is available. The novelty of our analysis is that the error estimates of all the unknowns are robust for material parameters. Specifically, the analysis does not need a uniformly positive storage coefficient, and the error estimates are robust for nearly incompressible materials. Numerical experiments illustrating our theoretical analysis are included.

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arXiv2016

Parameter-robust discretization and preconditioning of Biot's consolidation model

Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters ranges over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems. The approach taken here is to consider the stability of the problem in non-standard or weighted Hilbert spaces and employ the operator preconditioning approach. We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement. The parameters of interest are small time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.

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arXiv2015

Towards a unified analysis of mixed methods for elasticity with weakly symmetric stress

We propose a framework for unified analysis of mixed methods for elasticity with weakly symmetric stress. Based on a commuting diagram in the weakly symmetric elasticity complex and extending a previous stability result, stable mixed methods are obtained by combining Stokes stable and elasticity stable finite elements. We show that the framework can be used to analyze most existing mixed methods for the elasticity problem with elementary techniques. We also show that some new stable mixed finite elements are obtained.

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arXiv2014

Guaranteed locking-free finite element methods for Biot's consolidation model in poroelasticity

We propose a new finite element method for the three-field formulation of Biot's consolidation model in poroelasticity and prove a priori error estimates. Uniform-in-time error estimates of all the unknowns are obtained for both semidiscrete solutions and fully discrete solutions with the backward Euler time discretization. The novelty of our method is that the error analysis does not require the assumption that the constrained specific storage coefficient is uniformly positive. Therefore the method is guaranteed to be locking-free without the additional assumption on material parameters.

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arXiv2013

Mixed methods for elastodynamics with weak symmetry

We analyze the application to elastodynamic problems of mixed finite element methods for elasticity with weak symmetry. Our approach leads to a semidiscrete method which consists of a system of ordinary differential equations without algebraic constraints. Our error analysis, which is based on a new elliptic projection operator, applies to several mixed finite element spaces developed for elastostatics. The error estimates we obtain are robust for nearly incompressible materials.

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