28 research outputs found
Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods
In this article, we prove convergence of the weakly penalized adaptive
discontinuous Galerkin methods. Unlike other works, we derive the contraction
property for various discontinuous Galerkin methods only assuming the
stabilizing parameters are large enough to stabilize the method. A central idea
in the analysis is to construct an auxiliary solution from the discontinuous
Galerkin solution by a simple post processing. Based on the auxiliary solution,
we define the adaptive algorithm which guides to the convergence of adaptive
discontinuous Galerkin methods
Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem
In this paper, we analyze the convergence and optimality of a standard
adaptive nonconforming linear element method for the Stokes problem. After
establishing a special quasi--orthogonality property for both the velocity and
the pressure in this saddle point problem, we introduce a new prolongation
operator to carry through the discrete reliability analysis for the error
estimator. We then use a specially defined interpolation operator to prove
that, up to oscillation, the error can be bounded by the approximation error
within a properly defined nonlinear approximate class. Finally, by introducing
a new parameter-dependent error estimator, we prove the convergence and
optimality estimates
Instance optimal Crouzeix-Raviart adaptive finite element methods for the Poisson and Stokes problems
We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality
of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming
approximations of the Poisson problem to nonconforming Crouzeix-Raviart
approximations of the Poisson and the Stokes problem in 2D. As a consequence,
we obtain instance optimality of an AFEM with a modified maximum marking
strategy
Numerical Simulations of Bouncing Jets
Bouncing jets are fascinating phenomenons occurring under certain conditions
when a jet impinges on a free surface. This effect is observed when the fluid
is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs
also for non-Newtonian fluids when the jets falls in a vessel at rest
containing the same fluid.
We investigate numerically the impact of the experimental setting and the
rheological properties of the fluid on the onset of the bouncing phenomenon.
Our investigations show that the occurrence of a thin lubricating layer of air
separating the jet and the rest of the liquid is a key factor for the bouncing
of the jet to happen.
The numerical technique that is used consists of a projection method for the
Navier-Stokes system coupled with a level set formulation for the
representation of the interface. The space approximation is done with adaptive
finite elements. Adaptive refinement is shown to be very important to capture
the thin layer of air that is responsible for the bouncing
Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems
We prove that for compactly perturbed elliptic problems, where the
corresponding bilinear form satisfies a Garding inequality, adaptive
mesh-refinement is capable of overcoming the preasymptotic behavior and
eventually leads to convergence with optimal algebraic rates. As an important
consequence of our analysis, one does not have to deal with the a-priori
assumption that the underlying meshes are sufficiently fine. Hence, the overall
conclusion of our results is that adaptivity has stabilizing effects and can
overcome possibly pessimistic restrictions on the meshes. In particular, our
analysis covers adaptive mesh-refinement for the finite element discretization
of the Helmholtz equation from where our interest originated
Convergence and Optimality of Higher-Order Adaptive Finite Element Methods for Eigenvalue Clusters
Proofs of convergence of adaptive finite element methods for the
approximation of eigenvalues and eigenfunctions of linear elliptic problems
have been given in a several recent papers. A key step in establishing such
results for multiple and clustered eigenvalues was provided by Dai et. al.
(2014), who proved convergence and optimality of AFEM for eigenvalues of
multiplicity greater than one. There it was shown that a theoretical
(non-computable) error estimator for which standard convergence proofs apply is
equivalent to a standard computable estimator on sufficiently fine grids.
Gallistl (2015) used a similar tool in order to prove that a standard adaptive
FEM for controlling eigenvalue clusters for the Laplacian using continuous
piecewise linear finite element spaces converges with optimal rate. When
considering either higher-order finite element spaces or non-constant diffusion
coefficients, however, the arguments of Dai et. al. and Gallistl do not yield
equivalence of the practical and theoretical estimators for clustered
eigenvalues. In this note we provide this missing key step, thus showing that
standard adaptive FEM for clustered eigenvalues employing elements of arbitrary
polynomial degree converge with optimal rate. We additionally establish that a
key user-defined input parameter in the AFEM, the bulk marking parameter, may
be chosen entirely independently of the properties of the target eigenvalue
cluster. All of these results assume a fineness condition on the initial mesh
in order to ensure that the nonlinearity is sufficiently resolved.Comment: 10 page