832 research outputs found
Error analysis of a fully discrete Morley finite element approximation for the Cahn-Hilliard equation
This paper proposes and analyzes the Morley element method for the
Cahn-Hilliard equation. It is a fourth order nonlinear singular perturbation
equation arises from the binary alloy problem in materials science, and its
limit is proved to approach the Hele-Shaw flow. If the error
estimate is considered directly as in paper \cite{elliott1989nonconforming}, we
can only prove that the error bound depends on the exponential function of
. Instead, this paper derives the error bound which depends
on the polynomial function of by considering the discrete
error estimate first. There are two main difficulties in proving this
polynomial dependence of the discrete error estimate. Firstly, it is
difficult to prove discrete energy law and discrete stability results due to
the complex structure of the bilinear form of the Morley element
discretization. This paper overcomes this difficulty by defining four types of
discrete inverse Laplace operators and exploring the relations between these
discrete inverse Laplace operators and continuous inverse Laplace operator.
Each of these operators plays important roles, and their relations are crucial
in proving the discrete energy law, discrete stability results and error
estimates. Secondly, it is difficult to prove the discrete spectrum estimate in
the Morley element space because the Morley element space intersects with the
conforming finite element space but they are not contained in each other.
Instead of proving this discrete spectrum estimate in the Morley element space,
this paper proves a generalized coercivity result by exploring properties of
the enriching operators and using the discrete spectrum estimate in its
conforming relative finite element space, which can be obtained by using the
spectrum estimate of the Cahn-Hilliard operator.Comment: 31 page
Analysis of interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow
This paper develops and analyzes two fully discrete interior penalty
discontinuous Galerkin (IP-DG) methods for the Allen-Cahn equation, which is a
nonlinear singular perturbation of the heat equation and originally arises from
phase transition of binary alloys in materials science, and its sharp interface
limit (the mean curvature flow) as the perturbation parameter tends to zero.
Both fully implicit and energy-splitting time-stepping schemes are proposed.
The primary goal of the paper is to derive sharp error bounds which depend on
the reciprocal of the perturbation parameter (also called
"interaction length") only in some lower polynomial order, instead of
exponential order, for the proposed IP-DG methods. The derivation is based on a
refinement of the nonstandard error analysis technique first introduced in
[12]. The centerpiece of this new technique is to establish a spectrum estimate
result in totally discontinuous DG finite element spaces with a help of a
similar spectrum estimate result in the conforming finite element spaces which
was established in [12]. As a nontrivial application of the sharp error
estimates, they are used to establish convergence and the rates of convergence
of the zero level sets of the fully discrete IP-DG solutions to the classical
and generalized mean curvature flow. Numerical experiment results are also
presented to gauge the theoretical results and the performance of the proposed
fully discrete IP-DG methods.Comment: 24 pages, 2 tables and 12 graphic
Analysis of adaptive two-grid finite element algorithms for linear and nonlinear problems
This paper proposes some efficient and accurate adaptive two-grid (ATG)
finite element algorithms for linear and nonlinear partial differential
equations (PDEs). The main idea of these algorithms is to utilize the solutions
on the -th level adaptive meshes to find the solutions on the -th
level adaptive meshes which are constructed by performing adaptive element
bisections on the -th level adaptive meshes. These algorithms transform
non-symmetric positive definite (non-SPD) PDEs (resp., nonlinear PDEs) into
symmetric positive definite (SPD) PDEs (resp., linear PDEs). The proposed
algorithms are both accurate and efficient due to the following advantages:
they do not need to solve the non-symmetric or nonlinear systems; the degrees
of freedom (d.o.f.) are very small; they are easily implemented; the
interpolation errors are very small. Next, this paper constructs residue-type
{\em a posteriori} error estimators, which are shown to be reliable and
efficient. The key ingredient in proving the efficiency is to establish an
upper bound of the oscillation terms, which may not be higher-order terms
(h.o.t.) due to the low regularity of the numerical solution. Furthermore, the
convergence of the algorithms is proved when bisection is used for the mesh
refinements. Finally, numerical experiments are provided to verify the accuracy
and efficiency of the ATG finite element algorithms, compared to regular
adaptive finite element algorithms and two-grid finite element algorithms [27].Comment: 29 page
Fully Discrete Mixed Finite Element Methods for the Stochastic Cahn-Hilliard Equation with Gradient-type Multiplicative Noise
This paper develops and analyzes some fully discrete mixed finite element
methods for the stochastic Cahn-Hilliard equation with gradient-type
multiplicative noise that is white in time and correlated in space. The
stochastic Cahn-Hilliard equation is formally derived as a phase field
formulation of the stochastically perturbed Hele-Shaw flow. The main result of
this paper is to prove strong convergence with optimal rates for the proposed
mixed finite element methods. To overcome the difficulty caused by the low
regularity in time of the solution to the stochastic Cahn-Hilliard equation,
the H\"{o}lder continuity in time with respect to various norms for the
stochastic PDE solution is established, and it plays a crucial role in the
error analysis. Numerical experiments are also provided to validate the
theoretical results and to study the impact of noise on the Hele-Shaw flow as
well as the interplay of the geometric evolution and gradient-type noise.Comment: 22 pages, 6 figures and 2 table
Finite Element Methods for the Stochastic Allen-Cahn Equation with Gradient-type Multiplicative Noises
This paper studies finite element approximations of the stochastic Allen-Cahn
equation with gradient-type multiplicative noises that are white in time and
correlated in space. The sharp interface limit as the parameter of the stochastic equation formally approximates a stochastic
mean curvature flow which is described by a stochastically perturbed geometric
law of the deterministic mean curvature flow. Both the stochastic Allen-Cahn
equation and the stochastic mean curvature flow arise from materials science,
fluid mechanics and cell biology applications. Two fully discrete finite
element methods which are based on different time-stepping strategies for the
nonlinear term are proposed. Strong convergence with sharp rates for both fully
discrete finite element methods is proved. This is done with a crucial help of
the H\"{o}lder continuity in time with respect to the spatial -norm and
-seminorm for the strong solution of the stochastic Allen-Cahn equation,
which are key technical lemmas proved in paper. It also relies on the fact that
high moments of the strong solution are bounded in various spatial and temporal
norms. Numerical experiments are provided to gauge the performance of the
proposed fully discrete finite element methods and to study the interplay of
the geometric evolution and gradient-type noises.Comment: 28 pages, 8 figures, 1 tabl
Multiphysics Finite Element Methods for a Poroelasticity Model
This paper concerns with finite element approximations of a quasi-static
poroelasticity model in displacement-pressure formulation which describes the
dynamics of poro-elastic materials under an applied mechanical force on the
boundary. To better describe the multiphysics process of deformation and
diffusion for poro-elastic materials, we first present a reformulation of the
original model by introducing two pseudo-pressures, one of them is shown to
satisfy a diffusion equation, we then propose a time-stepping algorithm which
decouples (or couples) the reformulated PDE problem at each time step into two
sub-problems, one of which is a generalized Stokes problem for the displacement
vector field (of the solid network of the poro-elastic material) along with one
pseudo-pressure field and the other is a diffusion problem for the other
pseudo-pressure field (of the solvent of the material). In the paper, the
Taylor-Hood mixed finite element method combined with the -conforming
finite element method is used as an example to demonstrate the viability of the
proposed multiphysics approach. It is proved that the solutions of the fully
discrete finite element methods fulfill a discrete energy law which mimics the
differential energy law satisfied by the PDE solution and converges optimally
in the energy norm. Moreover, it is showed that the proposed formulation also
has a built-in mechanism to overcome so-called "locking phenomenon" associated
with the numerical approximations of the poroelasticity model. Numerical
experiments are presented to show the performance of the proposed approach and
methods and to demonstrate the absence of "locking phenomenon" in our numerical
experiments.Comment: 8 figures and 1 tabl
Kozai-Lidov Mechanism inside Retrograde Mean Motion Resonances
As the discoveries of more minor bodies in retrograde resonances with giant
planets, such as 2015 BZ509 and 2006 RJ2, our curiosity about the Kozai-Lidov
dynamics inside the retrograde resonance has been sparked. In this study, we
focus on the 3D retrograde resonance problem and investigate how the resonant
dynamics of a minor body impacts on its own Kozai-Lidov cycle. Firstly we
deduce the action-angle variables and canonical transformations that deal with
the retrograde orbit specifically. After obtaining the dominant Hamiltonian of
this problem, we then carry out the numerical averaging process in closed form
to generate phase-space portraits on a space. The retrograde 1:1
resonance is particularly scrutinized in detail, and numerical results from a
CRTBP model shows a great agreement with the our semi-analytical portraits. On
this basis, we inspect two real minor bodies currently trapped in retrograde
1:1 mean motion resonance. It is shown that they have different Kozai-Lidov
states, which can be used to analyze the stability of their unique resonances.
In the end, we further inspect the Kozai-Lidov dynamics inside the 2:1 and 2:5
retrograde resonance, and find distinct dynamical bifurcations of equilibrium
points on phase-space portraits.Comment: 10 pages, 8 figures. Accepted for publication in MNRA
Strong convergence of a fully discrete finite element method for a class of semilinear stochastic partial differential equations with multiplicative noise
This paper develops and analyzes a fully discrete finite element method for a
class of semilinear stochastic partial differential equations (SPDEs) with
multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is
assumed to be globally Lipschitz and the nonlinearity in the drift term is only
assumed to satisfy a one-side Lipschitz condition. The semilinear SPDEs
considered in this paper is a direct generalization of the SODEs considered in
[13]. There are several difficulties which need to be overcome for this
generalization. First, obviously the spatial discretization, which does not
appear in the SODE case, adds an extra layer of difficulty. It turns out a
special discretization must be designed to guarantee certain properties for the
numerical scheme and its stiffness matrix. In this paper we use a finite
element interpolation technique to discretize the nonlinear drift term. Second,
in order to prove the strong convergence of the proposed fully discrete finite
element method, stability estimates for higher order moments of the
-seminorm of the numerical solution must be established, which are
difficult and delicate. A judicious combination of the properties of the drift
and diffusion terms and a nontrivial technique borrowed from [16] is used in
this paper to achieve the goal. Finally, stability estimates for the second and
higher order moments of the -norm of the numerical solution is also
difficult to obtain due to the fact that the mass matrix may not be diagonally
dominant. This is done by utilizing the interpolation theory and the higher
moment estimates for the -seminorm of the numerical solution. After
overcoming these difficulties, it is proved that the proposed fully discrete
finite element method is convergent in strong norms with nearly optimal rates
of convergence.Comment: 23 pages. 11 figure
Analysis of mixed interior penalty discontinuous Galerkin methods for the Cahn-Hilliard equation and the Hele-Shaw flow
This paper proposes and analyzes two fully discrete mixed interior penalty
discontinuous Galerkin (DG) methods for the fourth order nonlinear
Cahn-Hilliard equation. Both methods use the backward Euler method for time
discretization and interior penalty discontinuous Galerkin methods for spatial
discretization. They differ from each other on how the nonlinear term is
treated, one of them is based on fully implicit time-stepping and the other
uses the energy-splitting time-stepping. The primary goal of the paper is to
prove the convergence of the numerical interfaces of the DG methods to the
interface of the Hele-Shaw flow. This is achieved by establishing error
estimates that depend on only in some low polynomial orders,
instead of exponential orders. Similar to [14], the crux is to prove a discrete
spectrum estimate in the discontinuous Galerkin finite element space. However,
the validity of such a result is not obvious because the DG space is not a
subspace of the (energy) space and it is larger than the finite element
space. This difficult is overcome by a delicate perturbation argument which
relies on the discrete spectrum estimate in the finite element space proved in
\cite{Feng_Prohl04}. Numerical experiment results are also presented to gauge
the theoretical results and the performance of the proposed fully discrete
mixed DG methods.Comment: 30 pages, 3 tables and 6 figures. arXiv admin note: text overlap with
arXiv:1310.750
Energy Conserving Galerkin Approximation of Two Dimensional Wave Equations with Random Coefficients
Wave propagation problems for heterogeneous media are known to have many
applications in physics and engineering. Recently, there has been an increasing
interest in stochastic effects due to the uncertainty, which may arise from
impurities of the media. This work considers a two-dimensional wave equation
with random coefficients which may be discontinuous in space. Generalized
polynomial chaos method is used in conjunction with stochastic Galerkin
approximation, and local discontinuous Galerkin method is used for spatial
discretization. Our method is shown to be energy preserving in semi-discrete
form as well as in fully discrete form, when leap-frog time discretization is
used. Its convergence rate is proved to be optimal and the error grows linearly
in time. The theoretical properties of the proposed scheme are validated by
numerical tests
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