271 research outputs found
Multigrid solvers for multipoint flux approximations of the Darcy problem on rough quadrilateral grids
In this work, an efficient blackbox-type multigrid method is proposed for solving multipoint flux approximations of the Darcy problem on logically rectangular grids. The approach is based on a cell-centered multigrid algorithm, which combines a piecewise constant interpolation and the restriction operator by Wesseling/Khalil with a line-wise relaxation procedure. A local Fourier analysis is performed for the case of a Cartesian uniform grid. The method shows a robust convergence for different full tensor coefficient problems and several rough quadrilateral grids.Francisco J. Gaspar has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska–Curie grant agreement no. 705402, POROSOS. The work of Laura Portero is supported by the Spanish project MTM2016-75139-R (AEI/FEDER, UE) and the Young Researchers Programme 2018 from the Public University of Navarre. Andrés Arrarás acknowledges support from the Spanish project PGC2018-099536-A-I00 (MCIU/AEI/FEDER, UE) and the Young Researchers Programme 2018 from the Public University of Navarre. The work of Carmen Rodrigo is supported by the Spanish project PGC2018-099536-A-I00 (MCIU/AEI/FEDER, UE) and the DGA (Grupo de referencia APEDIF, ref. E24_17R)
Raviart-Thomas finite elements of Petrov-Galerkin type
The mixed finite element method for the Poisson problem with the
Raviart-Thomas elements of low-level can be interpreted as a finite volume
method with a non-local gradient. In this contribution, we propose a variant of
Petrov-Galerkin type for this problem to ensure a local computation of the
gradient at the interfaces of the elements. The shape functions are the
Raviart-Thomas finite elements. Our goal is to define test functions that are
in duality with these shape functions: Precisely, the shape and test functions
will be asked to satisfy a L2-orthogonality property. The general theory of
Babu\v{s}ka brings necessary and sufficient stability conditions for a
Petrov-Galerkin mixed problem to be convergent. We propose specific constraints
for the dual test functions in order to ensure stability. With this choice, we
prove that the mixed Petrov-Galerkin scheme is identical to the four point
finite volumes scheme of Herbin, and to the mass lumping approach developed by
Baranger, Maitre and Oudin. Finally, we construct a family of dual test
functions that satisfy the stability conditions. Convergence is proven with the
usual techniques of mixed finite elements
Stable cell-centered finite volume discretization for Biot equations
In this paper we discuss a new discretization for the Biot equations. The
discretization treats the coupled system of deformation and flow directly, as
opposed to combining discretizations for the two separate sub-problems. The
coupled discretization has the following key properties, the combination of
which is novel: 1) The variables for the pressure and displacement are
co-located, and are as sparse as possible (e.g. one displacement vector and one
scalar pressure per cell center). 2) With locally computable restrictions on
grid types, the discretization is stable with respect to the limits of
incompressible fluid and small time-steps. 3) No artificial stabilization term
has been introduced. Furthermore, due to the finite volume structure embedded
in the discretization, explicit local expressions for both momentum-balancing
forces as well as mass-conservative fluid fluxes are available.
We prove stability of the proposed method with respect to all relevant
limits. Together with consistency, this proves convergence of the method.
Finally, we give numerical examples verifying both the analysis and convergence
of the method
A Comparison of Consistent Discretizations for Elliptic Problems on Polyhedral Grids
In this work, we review a set of consistent discretizations for second-order elliptic equations, and compare and contrast them with respect to accuracy, monotonicity, and factors affecting their computational cost (degrees of freedom, sparsity, and condition numbers). Our comparisons include the linear and nonlinear TPFA method, multipoint flux-approximation (MPFA-O), mimetic methods, and virtual element methods. We focus on incompressible flow and study the effects of deformed cell geometries and anisotropic permeability.acceptedVersio
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
- …