1,190 research outputs found
Unified convergence analysis of numerical schemes for a miscible displacement problem
This article performs a unified convergence analysis of a variety of
numerical methods for a model of the miscible displacement of one
incompressible fluid by another through a porous medium. The unified analysis
is enabled through the framework of the gradient discretisation method for
diffusion operators on generic grids. We use it to establish a novel
convergence result in of the approximate
concentration using minimal regularity assumptions on the solution to the
continuous problem. The convection term in the concentration equation is
discretised using a centred scheme. We present a variety of numerical tests
from the literature, as well as a novel analytical test case. The performance
of two schemes are compared on these tests; both are poor in the case of
variable viscosity, small diffusion and medium to small time steps. We show
that upstreaming is not a good option to recover stable and accurate solutions,
and we propose a correction to recover stable and accurate schemes for all time
steps and all ranges of diffusion
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
Non-negative mixed finite element formulations for a tensorial diffusion equation
We consider the tensorial diffusion equation, and address the discrete
maximum-minimum principle of mixed finite element formulations. In particular,
we address non-negative solutions (which is a special case of the
maximum-minimum principle) of mixed finite element formulations. The discrete
maximum-minimum principle is the discrete version of the maximum-minimum
principle.
In this paper we present two non-negative mixed finite element formulations
for tensorial diffusion equations based on constrained optimization techniques
(in particular, quadratic programming). These proposed mixed formulations
produce non-negative numerical solutions on arbitrary meshes for low-order
(i.e., linear, bilinear and trilinear) finite elements. The first formulation
is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative
constraint to the variational statement of the Raviart-Thomas formulation. The
second non-negative formulation based on the variational multiscale
formulation.
For the former formulation we comment on the affect of adding the
non-negative constraint on the local mass balance property of the
Raviart-Thomas formulation. We also study the performance of the active set
strategy for solving the resulting constrained optimization problems. The
overall performance of the proposed formulation is illustrated on three
canonical test problems.Comment: 40 pages using amsart style file, and 15 figure
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