Hybrid coupling of CG and HDG discretizations based on Nitsche’s method

This is a post-peer-review, pre-copyedit version of an article published in Computational mechanics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00466-019-01770-8A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal and elastic problems. The hybrid CG-HDG coupling exploits the definition of the numerical flux and the trace of the solution on the mesh faces to impose the transmission conditions between the CG and HDG subdomains. The con- tinuity of the solution is imposed in the CG problem via Nitsche’s method, whereas the equilibrium of the flux at the interface is naturally enforced as a Neumann con- dition in the HDG global problem. The proposed strategy does not affect the core structure of CG and HDG discretizations. In fact, the resulting formulation leads to a minimally-intrusive coupling, suitable to be integrated in existing CG and HDG libraries. Numerical experiments in two and three dimensions show optimal global convergence of the stress and superconvergence of the displacement field, locking-free approximation, as well as the potential to treat structural problems of engineering interest featuring multiple materials with compressible and nearly incompressible behaviors.Peer ReviewedPostprint (author's final draft

A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model

We study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is employed to impose weakly this condition. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. A series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the sensitivity of the model with respect to its parameters

EFFICIENT DISCRETIZATION TECHNIQUES AND DOMAIN DECOMPOSITION METHODS FOR POROELASTICITY

This thesis develops a new mixed finite element method for linear elasticity model with weakly enforced symmetry on simplicial and quadrilateral grids. Motivated by the multipoint flux mixed finite element method (MFMFE) for flow in porous media, the method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces and the trapezoidal (vertex) quadrature rule in order to localize the interaction of degrees of freedom. Particularly, this allows for local elimination of stress and rotation variables around each vertex and leads to a cell-centered system for the displacements. The stability analysis shows that the method is well-posed on simplicial and quadrilateral grids. Theoretical and numerical results indicate first-order convergence for all variables in the natural norms. Further discussion of the application of said Multipoint Stress Mixed Finite Element (MSMFE) method to the Biot system for poroelasticity is then presented. The flow part of the proposed model is treated in the MFMFE framework, while the mixed formulation for the elasticity equation is adopted for the use of the MSMFE technique. The extension of the MFMFE method to an arbitrary order finite volume scheme for solving elliptic problems on quadrilateral and hexahedral grids that reduce the underlying mixed finite element method to cell-centered pressure system is also discussed. A Multiscale Mortar Mixed Finite Element method for the linear elasticity on non-matching multiblock grids is also studied. A mortar finite element space is introduced on the nonmatching interfaces. In this mortar space the trace of the displacement is approximated, and continuity of normal stress is then weakly imposed. The condition number of the interface system is analyzed and optimal order of convergence is shown for stress, displacement, and rotation. Moreover, at cell centers, superconvergence is proven for the displacement variable. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory for all proposed approaches

Domain Decomposition And Time-Splitting Methods For The Biot System Of Poroelasticity

In this thesis, we develop efficient mixed finite element methods to solve the Biot system of poroelasticity, which models the flow of a viscous fluid through a porous medium along with the deformation of the medium. We study non-overlapping domain decomposition techniques and sequential splitting methods to reduce the computational complexity of the problem. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly the continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and split methods. For the monolithic method, the cases of matching and non-matching subdomain grid interfaces are analyzed separately. For both cases, a coupled displacement-pressure interface problem is solved, with each iteration requiring the solution of local Biot problems. For the case of matching subdomain grids, we show that the resulting interface operator is positive definite and analyze the convergence of the iteration. For the non-matching subdomain grid case, we use a multiscale mortar mixed finite element (MMMFE) approach. We further study drained split and fixed stress Biot splittings, in which case we solve separate interface problems requiring elasticity and Darcy solves. We analyze the stability of the split formulations. We also use numerical experiments to illustrate the convergence of the domain decomposition methods and compare their accuracy and efficiency in the monolithic and time-splitting settings. Finally, we present a novel space-time domain decomposition technique for the mixed finite element formulation of a parabolic equation. This method is motivated by the MMMFE method, where we split the space-time domain into multiple subdomains with space-time grids of different sizes. Scalar Lagrange multiplier (mortar) functions are introduced to enforce weakly the continuity of the normal component of the mixed finite element flux variable over the space-time interfaces. We analyze the new method and numerical experiments are developed to illustrate and confirm the theoretical results

The mortar boundary element method

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis is primarily concerned with the mortar boundary element method (mortar BEM). The mortar finite element method (mortar FEM) is a well established numerical scheme for the solution of partial differential equations. In simple terms the technique involves the splitting up of the domain of definition into separate parts. The problem may now be solved independently on these separate parts, however there must be some sort of matching condition between the separate parts. Our aim is to develop and analyse this technique to the boundary element method (BEM). The first step in our journey towards the mortar BEM is to investigate the BEM with Lagrangian multipliers. When approximating the solution of Neumann problems on open surfaces by the Galerkin BEM the appropriate boundary condition (along the boundary curve of the surface) can easily be included in the definition of the spaces used. However, we introduce a boundary element Galerkin BEM where we use a Lagrangian multiplier to incorporate the appropriate boundary condition in a weak sense. This is the first step in enabling us to understand the necessary matching conditions for a mortar type decomposition. We next formulate the mortar BEM for hypersingular integral equations representing the elliptic boundary value problem of the Laplace equation in three dimensions (with Neumann boundary condition). We prove almost quasi-optimal convergence of the scheme in broken Sobolev norms of order 1/2. Sub-domain decompositions can be geometrically non-conforming and meshes must be quasi-uniform only on sub-domains. We present numerical results which confirm and underline the theory presented concerning the BEM with Lagrangian multipliers and the mortar BEM. Finally we discuss the application of the mortaring technique to the hypersingular integral equation representing the equations of linear elasticity. Based on the assumption of ellipticity of the appearing bilinear form on a constrained space we prove the almost quasi-optimal convergence of the scheme

Modelling of flow and transport of non-Newtonian fluids interacting with poroelastic media

We propose and analyze a model for solving the coupled problem arising in the interaction of a free fluid with a poroelastic structure. The flow in the fluid region is described by Stokes equations and in the poroelastic medium by the quasi-static Biot model. The focus of the model is on the quasi- Newtonian fluids that exhibit a shear-thinning property. We establish existence and uniqueness of the solution for two alternative formulations of the proposed model. Then we establish and show the existence and uniqueness of the solution of semidiscrete continuous-in-time formulation. We present complete stability and error analysis, as well as results of numerical simulations showing optimal rates of convergence for all variables. After that, the modeling of a transport equation in a non-linear Biot-Stokes flow will be analyzed. We use discontinuous Galerkin method to solve the transport equation. Several numerical tests are presented illustrating theoretical results and the capabilities of the method

The mortar boundary element method

This thesis is primarily concerned with the mortar boundary element method (mortar BEM). The mortar finite element method (mortar FEM) is a well established numerical scheme for the solution of partial differential equations. In simple terms the technique involves the splitting up of the domain of definition into separate parts. The problem may now be solved independently on these separate parts, however there must be some sort of matching condition between the separate parts. Our aim is to develop and analyse this technique to the boundary element method (BEM). The first step in our journey towards the mortar BEM is to investigate the BEM with Lagrangian multipliers. When approximating the solution of Neumann problems on open surfaces by the Galerkin BEM the appropriate boundary condition (along the boundary curve of the surface) can easily be included in the definition of the spaces used. However, we introduce a boundary element Galerkin BEM where we use a Lagrangian multiplier to incorporate the appropriate boundary condition in a weak sense. This is the first step in enabling us to understand the necessary matching conditions for a mortar type decomposition. We next formulate the mortar BEM for hypersingular integral equations representing the elliptic boundary value problem of the Laplace equation in three dimensions (with Neumann boundary condition). We prove almost quasi-optimal convergence of the scheme in broken Sobolev norms of order 1/2. Sub-domain decompositions can be geometrically non-conforming and meshes must be quasi-uniform only on sub-domains. We present numerical results which confirm and underline the theory presented concerning the BEM with Lagrangian multipliers and the mortar BEM. Finally we discuss the application of the mortaring technique to the hypersingular integral equation representing the equations of linear elasticity. Based on the assumption of ellipticity of the appearing bilinear form on a constrained space we prove the almost quasi-optimal convergence of the scheme.EThOS - Electronic Theses Online ServiceGBUnited Kingdo