514 research outputs found
A mixed finite element method for Darcy’s equations with pressure dependent porosity
In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy’s equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose
dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuska-Brezzi theory and the Banach fixed point Theorem. In particular, given any integer k ≥ 0, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order k for the
velocity, piecewise polynomials of degree k for the pressure, and continuous piecewise polynomials of degree k+1 for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two
different methods for solving the discrete schemes. Next, we derive a reliable and efficient residualbased a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Cl´ement interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.Centro de Investigación en IngenierÃa Matemática (CI2MA), Universidad de ConcepciónUniversity of LausanneMinistry of Education, Youth and Sports of the Czech Republi
Convergence and Optimality of Adaptive Mixed Finite Element Methods
The convergence and optimality of adaptive mixed finite element methods for
the Poisson equation are established in this paper. The main difficulty for
mixed finite element methods is the lack of minimization principle and thus the
failure of orthogonality. A quasi-orthogonality property is proved using the
fact that the error is orthogonal to the divergence free subspace, while the
part of the error that is not divergence free can be bounded by the data
oscillation using a discrete stability result. This discrete stability result
is also used to get a localized discrete upper bound which is crucial for the
proof of the optimality of the adaptive approximation
The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements
For both the Poisson model problem and the Stokes problem in any dimension,
this paper proves that the enriched Crouzeix-Raviart elements are actually
identical to the first order Raviart-Thomas elements in the sense that they
produce the same discrete stresses. This result improves the previous result in
literature which, for two dimensions, states that the piecewise constant
projection of the stress by the first order Raviart-Thomas element is equal to
that by the Crouzeix-Raviart element. For the eigenvalue problem of Laplace
operator, this paper proves that the error of the enriched Crouzeix-Raviart
element is equivalent to that of the Raviart-Thomas element up to higher order
terms
Convergence of an adaptive mixed finite element method for general second order linear elliptic problems
The convergence of an adaptive mixed finite element method for general second
order linear elliptic problems defined on simply connected bounded polygonal
domains is analyzed in this paper. The main difficulties in the analysis are
posed by the non-symmetric and indefinite form of the problem along with the
lack of the orthogonality property in mixed finite element methods. The
important tools in the analysis are a posteriori error estimators,
quasi-orthogonality property and quasi-discrete reliability established using
representation formula for the lowest-order Raviart-Thomas solution in terms of
the Crouzeix-Raviart solution of the problem. An adaptive marking in each step
for the local refinement is based on the edge residual and volume residual
terms of the a posteriori estimator. Numerical experiments confirm the
theoretical analysis.Comment: 24 pages, 8 figure
Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity
A stress equilibration procedure for linear elasticity is proposed and
analyzed in this paper with emphasis on the behavior for (nearly)
incompressible materials. Based on the displacement-pressure approximation
computed with a stable finite element pair, it constructs an -conforming, weakly symmetric stress reconstruction. Our focus is
on the Taylor-Hood combination of continuous finite element spaces of
polynomial degrees and for the displacement and the pressure,
respectively. Our construction leads then to reconstructed stresses by
Raviart-Thomas elements of degree which are weakly symmetric in the sense
that its anti-symmetric part is zero tested against continuous piecewise
polynomial functions of degree . The computation is performed locally on a
set of vertex patches covering the computational domain in the spirit of
equilibration \cite{BraSch:08}. Due to the weak symmetry constraint, the local
problems need to satisfy consistency conditions associated with all rigid body
modes, in contrast to the case of Poisson's equation where only the constant
modes are involved. The resulting error estimator is shown to constitute a
guaranteed upper bound for the error with a constant that depends only on the
shape regularity of the triangulation. Local efficiency, uniformly in the
incompressible limit, is deduced from the upper bound by the residual error
estimator
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