4 research outputs found
Multiscale mortar mixed finite element methods for the Biot system of poroelasticity
We develop a mixed finite element domain decomposition method on non-matching
grids for the Biot system of poroelasticity. A displacement-pressure vector
mortar function is introduced on the interfaces and utilized as a Lagrange
multiplier to impose weakly continuity of normal stress and normal velocity.
The mortar space can be on a coarse scale, resulting in a multiscale
approximation. We establish existence, uniqueness, stability, and error
estimates for the semidiscrete continuous-in-time formulation under a suitable
condition on the richness of the mortar space. We further consider a
fully-discrete method based on the backward Euler time discretization and show
that the solution of the algebraic system at each time step can be reduced to
solving a positive definite interface problem for the composite mortar
variable. A multiscale stress-flux basis is constructed, which makes the number
of subdomain solves independent of the number of iterations required for the
interface problem, as well as the number of time steps. We present numerical
experiments verifying the theoretical results and illustrating the multiscale
capabilities of the method for a heterogeneous benchmark problem
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
A space-time multiscale mortar mixed finite element method for parabolic equations
We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with non-matching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability, as well as a priori error estimates for the spatial and temporal errors are established. A space-time non-overlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed. Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time