89 research outputs found
Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media
A conservative flux postprocessing algorithm is presented for both
steady-state and dynamic flow models. The postprocessed flux is shown to have
the same convergence order as the original flux. An arbitrary flux
approximation is projected into a conservative subspace by adding a piecewise
constant correction that is minimized in a weighted norm. The application
of a weighted norm appears to yield better results for heterogeneous media than
the standard norm which has been considered in earlier works. We also
study the effect of different flux calculations on the domain boundary. In
particular we consider the continuous Galerkin finite element method for
solving Darcy flow and couple it with a discontinuous Galerkin finite element
method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table
A Hybrid High-Order method for nonlinear elasticity
In this work we propose and analyze a novel Hybrid High-Order discretization
of a class of (linear and) nonlinear elasticity models in the small deformation
regime which are of common use in solid mechanics. The proposed method is valid
in two and three space dimensions, it supports general meshes including
polyhedral elements and nonmatching interfaces, enables arbitrary approximation
order, and the resolution cost can be reduced by statically condensing a large
subset of the unknowns for linearized versions of the problem. Additionally,
the method satisfies a local principle of virtual work inside each mesh
element, with interface tractions that obey the law of action and reaction. A
complete analysis covering very general stress-strain laws is carried out, and
optimal error estimates are proved. Extensive numerical validation on model
test problems is also provided on two types of nonlinear models.Comment: 29 pages, 7 figures, 4 table
Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling
International audienceWe 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 h , whereas the interfaces are partitioned by a mesh of much coarser size H , 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 H/h 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 strategy is proposed and numerically tested
A review of recent advances in discretization methods, a posteriori error analysis, and adaptive algorithms for numerical modeling in geosciences
International audienceTwo research subjects in geosciences which lately underwent significant progress are treated in this review. In the first part we focus on one key ingredient for the numerical approximation of the Darcy flow problem, namely the discretization of diffusion terms on general polygonal/polyhedral meshes. We present different schemes and discuss in detail their fundamental numerical properties such as stability, consistency, and robustness. The second part of the paper is devoted to error control and adaptivity for model geosciences problems. We present the available a posteriori estimates guaranteeing the maximal overall error and show how the different error components can be identified. These estimates are used to formulate adaptive stopping criteria for linear and nonlinear solvers, time step choice adjustment, and adaptive mesh refinement. Numerical experiments illustrate such entirely adaptive algorithms
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
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