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In this paper, we study the boundary penalty method for optimal control of unsteady Navier–Stokes type system that has been proposed as an alternative for Dirichlet boundary control. Existence and uniqueness of solutions are demonstrated and existence of optimal control for a class of optimal control problems is established. The asymptotic behavior of solution, with respect to the penalty parameter ϵ, is studied. In particular, we prove convergence of solutions of penalized control problem to the...
We present our work on the numerical solution of a continuum model of flocking dynamics in two spatial dimensions. The model consists of the compressible Euler equations with a nonlinear nonlocal term which requires special treatment. We use a semi-implicit discontinuous Galerkin scheme, which proves to be efficient enough to produce results in 2D in reasonable time. This work is a direct extension of the authors' previous work in 1D.
The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation...
We present a discontinuous Galerkin scheme for the numerical approximation of the
one-dimensional periodic Vlasov-Poisson equation. The scheme is based on a
Galerkin-characteristics method in which the distribution function is projected onto a
space of discontinuous functions. We present comparisons with a semi-Lagrangian method to
emphasize the good behavior of this scheme when applied to Vlasov-Poisson test cases.
In this paper we prove the discrete compactness property for
a discontinuous Galerkin approximation of Maxwell's system
on quite general tetrahedral meshes.
As a consequence, a discrete Friedrichs inequality is obtained
and the convergence of the discrete eigenvalues to the continuous ones is deduced
using the theory of collectively compact operators.
Some numerical experiments confirm the theoretical predictions.
Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods are
developed for solving the arising systems of convection-diffusion-dispersion-reaction
equations, and the received results of several discretization
methods are presented. We concentrate on linear reaction systems, which
can be solved analytically.
In the numerical methods, we use large time-steps to achieve
long simulation times of about 10 000 years.
We propose...
A mixed finite element method for the Navier–Stokes equations is introduced in which the stress is a primary variable. The variational formulation retains the mathematical structure of the Navier–Stokes equations and the classical theory extends naturally to this setting. Finite element spaces satisfying the associated inf–sup conditions are developed.
We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. The
membrane is characterized by its Canham-Helfrich energy (Willmore energy with area
constraint) and acts as a boundary force on the Navier-Stokes system modeling an
incompressible fluid. We give a concise description of the model and of the associated
numerical scheme. We provide numerical simulations with emphasis on the comparisons
between different types of flow:...
We introduce a new way of computation of time dependent partial differential equations using hybrid method FEM in space and FDM in time domain and explicit computational scheme. The key idea is quick transformation of standard basis functions into new simple basis functions. This new way is used for better computational efficiency. We explain this way of computation on an example of elastodynamic equation using quadrilateral elements. However, the method can be used for more types of elements and...
We introduce a new efficient way of computation of partial differential equations using a hybrid method composed from FEM in space and FDM in time domain. The overall computational scheme is explicit in time. The key idea of the suggested way is based on a transformation of standard basis functions into new basis functions. The results of this matrix transformation are e-invariants (effective invariants) with such suitable properties which save the number of arithmetical operations needed for a...
In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter
dependence to problems involving (a) nonaffine dependence on the
parameter, and (b) nonlinear dependence on the field variable.
The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational
decomposition. We first review the coefficient function...
We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatly improves accuracy in simulations. Standard finite element schemes for NS-α suffer from two major sources of error if their solutions are considered approximations to true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure error on the...
We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatly
improves accuracy in simulations. Standard finite element schemes
for NS-α suffer from two major sources of error if their solutions are considered approximations
to true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure error
on the...
We rigorously derive energy estimates for the second order vector wave equation with gauge condition for the electric field with non-constant electric permittivity function. This equation is used in the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system. Our numerical experiments illustrate efficiency of the modified hybrid scheme in two and three space dimensions when the method is applied for generation of backscattering data in the reconstruction...
We consider the lowest-order Raviart–Thomas mixed finite element
method for second-order elliptic problems on simplicial meshes in
two and three space dimensions. This method produces saddle-point
problems for scalar and flux unknowns. We show how to easily and
locally eliminate the flux unknowns, which implies the equivalence
between this method and a particular multi-point finite volume
scheme, without any approximate numerical integration. The matrix
of the final linear system is sparse, positive...
The phase relaxation model is a diffuse interface model with
small parameter ε which
consists of a parabolic PDE for temperature
θ and an ODE with double obstacles
for phase variable χ.
To decouple the system a semi-explicit Euler method with variable
step-size τ is used for time discretization, which requires
the stability constraint τ ≤ ε. Conforming piecewise
linear finite elements over highly graded simplicial meshes
with parameter h are further employed for space discretization.
A posteriori...
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