1,474 research outputs found
Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations
This paper is concerned with the discretization error analysis of semilinear
Neumann boundary control problems in polygonal domains with pointwise
inequality constraints on the control. The approximations of the control are
piecewise constant functions. The state and adjoint state are discretized by
piecewise linear finite elements. In a postprocessing step approximations of
locally optimal controls of the continuous optimal control problem are
constructed by the projection of the respective discrete adjoint state.
Although the quality of the approximations is in general affected by corner
singularities a convergence order of is proven for domains
with interior angles smaller than using quasi-uniform meshes. For
larger interior angles mesh grading techniques are used to get the same order
of convergence
Numerical analysis for the pure Neumann control problem using the gradient discretisation method
The article discusses the gradient discretisation method (GDM) for
distributed optimal control problems governed by diffusion equation with pure
Neumann boundary condition. Using the GDM framework enables to develop an
analysis that directly applies to a wide range of numerical schemes, from
conforming and non-conforming finite elements, to mixed finite elements, to
finite volumes and mimetic finite differences methods. Optimal order error
estimates for state, adjoint and control variables for low order schemes are
derived under standard regularity assumptions. A novel projection relation
between the optimal control and the adjoint variable allows the proof of a
super-convergence result for post-processed control. Numerical experiments
performed using a modified active set strategy algorithm for conforming,
nonconforming and mimetic finite difference methods confirm the theoretical
rates of convergence
A linear domain decomposition method for partially saturated flow in porous media
The Richards equation is a nonlinear parabolic equation that is commonly used
for modelling saturated/unsaturated flow in porous media. We assume that the
medium occupies a bounded Lipschitz domain partitioned into two disjoint
subdomains separated by a fixed interface . This leads to two problems
defined on the subdomains which are coupled through conditions expressing flux
and pressure continuity at . After an Euler implicit discretisation of
the resulting nonlinear subproblems a linear iterative (-type) domain
decomposition scheme is proposed. The convergence of the scheme is proved
rigorously. In the last part we present numerical results that are in line with
the theoretical finding, in particular the unconditional convergence of the
scheme. We further compare the scheme to other approaches not making use of a
domain decomposition. Namely, we compare to a Newton and a Picard scheme. We
show that the proposed scheme is more stable than the Newton scheme while
remaining comparable in computational time, even if no parallelisation is being
adopted. Finally we present a parametric study that can be used to optimize the
proposed scheme.Comment: 34 pages, 13 figures, 7 table
A reduced basis localized orthogonal decomposition
In this work we combine the framework of the Reduced Basis method (RB) with
the framework of the Localized Orthogonal Decomposition (LOD) in order to solve
parametrized elliptic multiscale problems. The idea of the LOD is to split a
high dimensional Finite Element space into a low dimensional space with
comparably good approximation properties and a remainder space with negligible
information. The low dimensional space is spanned by locally supported basis
functions associated with the node of a coarse mesh obtained by solving
decoupled local problems. However, for parameter dependent multiscale problems,
the local basis has to be computed repeatedly for each choice of the parameter.
To overcome this issue, we propose an RB approach to compute in an "offline"
stage LOD for suitable representative parameters. The online solution of the
multiscale problems can then be obtained in a coarse space (thanks to the LOD
decomposition) and for an arbitrary value of the parameters (thanks to a
suitable "interpolation" of the selected RB). The online RB-LOD has a basis
with local support and leads to sparse systems. Applications of the strategy to
both linear and nonlinear problems are given
Coupling of cytoplasm and adhesion dynamics determines cell polarization and locomotion
Observations of single epidermal cells on flat adhesive substrates have
revealed two distinct morphological and functional states, namely a
non-migrating symmetric unpolarized state and a migrating asymmetric polarized
state. These states are characterized by different spatial distributions and
dynamics of important biochemical cell components: F-actin and myosin-II form
the contractile part of the cytoskeleton, and integrin receptors in the plasma
membrane connect F-actin filaments to the substratum. In this way, focal
adhesion complexes are assembled, which determine cytoskeletal force
transduction and subsequent cell locomotion. So far, physical models have
reduced this phenomenon either to gradients in regulatory control molecules or
to different mechanics of the actin filament system in different regions of the
cell.
Here we offer an alternative and self-organizational model incorporating
polymerization, pushing and sliding of filaments, as well as formation of
adhesion sites and their force dependent kinetics. All these phenomena can be
combined into a non-linearly coupled system of hyperbolic, parabolic and
elliptic differential equations. Aim of this article is to show how relatively
simple relations for the small-scale mechanics and kinetics of participating
molecules may reproduce the emergent behavior of polarization and migration on
the large-scale cell level.Comment: v2 (updates from proof): add TOC, clarify Fig. 4, fix several typo
A localized orthogonal decomposition method for semi-linear elliptic problems
In this paper we propose and analyze a new Multiscale Method for solving
semi-linear elliptic problems with heterogeneous and highly variable
coefficient functions. For this purpose we construct a generalized finite
element basis that spans a low dimensional multiscale space. The basis is
assembled by performing localized linear fine-scale computations in small
patches that have a diameter of order H |log H| where H is the coarse mesh
size. Without any assumptions on the type of the oscillations in the
coefficients, we give a rigorous proof for a linear convergence of the H1-error
with respect to the coarse mesh size. To solve the arising equations, we
propose an algorithm that is based on a damped Newton scheme in the multiscale
space
A Neumann interface optimal control problem with elliptic PDE constraints and its discretization and numerical analysis
We study an optimal control problem governed by elliptic PDEs with interface,
which the control acts on the interface. Due to the jump of the coefficient
across the interface and the control acting on the interface, the regularity of
solution of the control problem is limited on the whole domain, but smoother on
subdomains. The control function with pointwise inequality constraints is
served as the flux jump condition which we called Neumann interface control. We
use a simple uniform mesh that is independent of the interface. The standard
linear finite element method can not achieve optimal convergence when the
uniform mesh is used. Therefore the state and adjoint state equations are
discretized by piecewise linear immersed finite element method (IFEM). While
the accuracy of the piecewise constant approximation of the optimal control on
the interface is improved by a postprocessing step which possesses
superconvergence properties; as well as the variational discretization concept
for the optimal control is used to improve the error estimates. Optimal error
estimates for the control, suboptimal error estimates for state and adjoint
state are derived. Numerical examples with and without constraints are provided
to illustrate the effectiveness of the proposed scheme and correctness of the
theoretical analysis.Comment: 31pages, 12 figures, 4 table
- …