694 research outputs found
Weak solutions of the Landau--Lifshitz--Bloch equation
The Landau--Lifshitz--Bloch (LLB) equation is a formulation of dynamic
micromagnetics valid at all temperatures, treating both the transverse and
longitudinal relaxation components important for high-temperature applications.
We study LLB equation in case the temperature raised higher than the Curie
temperature. The existence of weak solution is showed and its regularity
properties are also discussed. In this way, we lay foundations for the rigorous
theory of LLB equation that is currently not available
The gradient discretisation method for slow and fast diffusion porous media equations
The gradient discretisation method (GDM) is a generic framework for designing
and analysing numerical schemes for diffusion models. In this paper, we study
the GDM for the porous medium equation, including fast diffusion and slow
diffusion models, and a concentration-dependent diffusion tensor. Using
discrete functional analysis techniques, we establish a strong
-convergence of the approximate gradients and a uniform-in-time
convergence for the approximate solution, without assuming non-physical
regularity assumptions on the data or continuous solution. Being established in
the generic GDM framework, these results apply to a variety of numerical
methods, such as finite volume, (mass-lumped) finite elements, etc. The
theoretical results are illustrated, in both fast and slow diffusion regimes,
by numerical tests based on two methods that fit the GDM framework: mass-lumped
conforming finite elements and the Hybrid Mimetic Mixed method
A finite element approximation for the stochastic Landau-Lifshitz-Gilbert equation
The stochastic Landau--Lifshitz--Gilbert (LLG) equation describes the
behaviour of the magnetization under the influence of the effective field
consisting of random fluctuations. We first reformulate the equation into an
equation the unknown of which is differentiable with respect to the time
variable. We then propose a convergent -linear scheme for the numerical
solution of the reformulated equation. As a consequence, we show the existence
of weak martingale solutions to the stochastic LLG equation. A salient feature
of this scheme is that it does not involve a nonlinear system, and that no
condition on time and space steps is required when .
Numerical results are presented to show the applicability of the method
A semidiscrete finite element approximation of a time-fractional Fokker-Planck equation with nonsmooth initial data
We present a new stability and convergence analysis for the spatial
discretization of a time-fractional Fokker--Planck equation in a convex
polyhedral domain, using continuous, piecewise-linear, finite elements. The
forcing may depend on time as well as on the spatial variables, and the initial
data may have low regularity. Our analysis uses a novel sequence of energy
arguments in combination with a generalized Gronwall inequality. Although this
theory covers only the spatial discretization, we present numerical experiments
with a fully discrete scheme employing a very small time step, and observe
results consistent with the predicted convergence behavior
Finite element approximation of a time-fractional diffusion problem in a non-convex polygonal domain
An initial-boundary value problem for the time-fractional diffusion equation
is discretized in space using continuous piecewise-linear finite elements on a
polygonal domain with a re-entrant corner. Known error bounds for the case of a
convex polygon break down because the associated Poisson equation is no longer
-regular. In particular, the method is no longer second-order accurate if
quasi-uniform triangulations are used. We prove that a suitable local mesh
refinement about the re-entrant corner restores second-order convergence. In
this way, we generalize known results for the classical heat equation due to
Chatzipantelidis, Lazarov, Thom\'ee and Wahlbin.Comment: 21 pages, 4 figure
Numerical solution of the time-fractional Fokker-Planck equation with general forcing
We study two schemes for a time-fractional Fokker-Planck equation with space-
and time-dependent forcing in one space dimension. The first scheme is
continuous in time and is discretized in space using a piecewise-linear
Galerkin finite element method. The second is continuous in space and employs a
time-stepping procedure similar to the classical implicit Euler method. We show
that the space discretization is second-order accurate in the spatial
-norm, uniformly in time, whereas the corresponding error for the
time-stepping scheme is for a uniform time step , where
is the fractional diffusion parameter. In numerical
experiments using a combined, fully-discrete method, we observe convergence
behaviour consistent with these results.Comment: 3 Figure
A finite element approximation for the stochastic Landau--Lifshitz--Gilbert equation with multi-dimensional noise
We propose an unconditionally convergent linear finite element scheme for the
stochastic Landau--Lifshitz--Gilbert (LLG) equation with multi-dimensional
noise. By using the Doss-Sussmann technique, we first transform the stochastic
LLG equation into a partial differential equation that depends on the solution
of the auxiliary equation for the diffusion part. The resulting equation has
solutions absolutely continuous with respect to time. We then propose a
convergent -linear scheme for the numerical solution of the
reformulated equation. As a consequence, we are able to show the existence of
weak martingale solutions to the stochastic LLG equation
Deep Recurrent Level Set for Segmenting Brain Tumors
Variational Level Set (VLS) has been a widely used method in medical
segmentation. However, segmentation accuracy in the VLS method dramatically
decreases when dealing with intervening factors such as lighting, shadows,
colors, etc. Additionally, results are quite sensitive to initial settings and
are highly dependent on the number of iterations. In order to address these
limitations, the proposed method incorporates VLS into deep learning by
defining a novel end-to-end trainable model called as Deep Recurrent Level Set
(DRLS). The proposed DRLS consists of three layers, i.e, Convolutional layers,
Deconvolutional layers with skip connections and LevelSet layers. Brain tumor
segmentation is taken as an instant to illustrate the performance of the
proposed DRLS. Convolutional layer learns visual representation of brain tumor
at different scales. Since brain tumors occupy a small portion of the image,
deconvolutional layers are designed with skip connections to obtain a high
quality feature map. Level-Set Layer drives the contour towards the brain
tumor. In each step, the Convolutional Layer is fed with the LevelSet map to
obtain a brain tumor feature map. This in turn serves as input for the LevelSet
layer in the next step. The experimental results have been obtained on
BRATS2013, BRATS2015 and BRATS2017 datasets. The proposed DRLS model improves
both computational time and segmentation accuracy when compared to the the
classic VLS-based method. Additionally, a fully end-to-end system DRLS achieves
state-of-the-art segmentation on brain tumors
Convergence analysis of a family of ELLAM schemes for a fully coupled model of miscible displacement in porous media
We analyse the convergence of numerical schemes in the GDM-ELLAM (Gradient
Discretisation Method-Eulerian Lagrangian Localised Adjoint Method) framework
for a strongly coupled elliptic-parabolic PDE which models miscible
displacement in porous media. These schemes include, but are not limited to
Mixed Finite Element-ELLAM and Hybrid Mimetic Mixed-ELLAM schemes. A complete
convergence analysis is presented on the coupled model, using only weak
regularity assumptions on the solution (which are satisfied in practical
applications), and not relying on bounds (which are impossible to
ensure at the discrete level given the anisotropic diffusion tensors and the
general grids used in applications)
A combined GDM--ELLAM--MMOC scheme for advection dominated PDEs
We propose a combination of the Eulerian Lagrangian Localised Adjoint Method
(ELLAM) and the Modified Method of Characteristics (MMOC) for time-dependent
advection-domina\-ted PDEs. The combined scheme, so-called GEM scheme, takes
advantages of both ELLAM scheme (mass conservation) and MMOC scheme (easier
computations), while at the same time avoids their disadvantages (respectively,
harder tracking around the injection regions, and loss of mass).
We present a precise analysis of mass conservation properties for these three
schemes, and after achieving global mass balance, an adjustment yielding local
volume conservation is then proposed. Numerical results for all three schemes
are then compared, illustrating the advantages of the GEM scheme.
A convergence result of the MMOC scheme, motivated by our previous work on
the convergence of ELLAM schemes, is provided, which can be extended to obtain
the convergence of GEM scheme
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