Gergina Pencheva

A MULTISCALE MORTAR MIXED FINITE ELEMENT METHOD

We develop multiscale mortar mixed finite element discretizations for second order elliptic equations. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. The polynomial degree of the mortar and subdomain approximation spaces may differ; in fact, the mortar space achieves approximation comparable to the fine scale on its coarse grid by using higher order polynomials. Our formulation is related to, but more flexible than, existing multiscale finite element and variational multiscale methods. We derive a priori error estimates and show, with appropriate choice of the mortar space, optimal order convergence and some superconvergence on the fine scale for both the solution and its flux. We also derive efficient and reliable a posteriori error estimators, which are used in an adaptive mesh refinement algorithm to obtain appropriate subdomain and mortar grids. Numerical experi...

Balancing domain decomposition for mortar mixed finite element methods

The balancing domain decomposition method for mixed finite elements by Cowsar, Mandel, and Wheeler is extended to the case of mortar mixed finite elements on non-matching multiblock grids. The algorithm involves an iterative solution of a mortar interface problem with one local Dirichlet solve and one local Neumann solve per subdomain on each iteration. A coarse solve is used to guarantee that the Neumann problems are consistent and to provide global exchange of information across subdomains. Quasi-optimal condition number bounds are derived, which are independent of the jump in coefficients between subdomains. Numerical experiments confirm the theoretical results. Copyright

DOMAIN DECOMPOSITION FOR POROELASTICITY AND ELASTICITY WITH DG JUMPS AND MORTARS

We couple a time-dependent poroelastic model in a region with an elastic model in adjacent regions. We discretize each model independently on non-matching grids and we realize a domain decomposition on the interface between the regions by introducing DG jumps and mortars. The unknowns are condensed on the interface, so that at each time step, the computation in each subdomain can be performed in parallel. In addition, by extrapolating the displacement, we present an algorithm where the computations of the pressure and displacement are decoupled. We show that the matrix of the interface problem is positive definite and establish error estimates for this scheme.

Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling

We consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size

A Frozen Jacobian Multiscale Mortar Preconditioner for Nonlinear Interface Operators

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A Global Jacobian Method for Mortar Discretizations of a Fully Implicit Two-Phase Flow Model

, whereas the interfaces are partitioned by a mesh of much coarser size

A global jacobian method for mortar discretizations of nonlinear porous media flows

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Discontinuous and Enriched Galerkin Methods for Phase-Field Fracture Propagation in Elasticity

, and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio

A multiscale mortar method and two-stage preconditioner for multiphase flow using a global jacobian approach

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An approximate Jacobian nonlinear solver for multiphase flow and transport

under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement s...