In this paper, we study adaptive finite element approximation in the backward Euler scheme for a constrained optimal control problem by parabolic equations on multi-meshes. The control constraint is given in an integral sense: $K = \{u(t) ∈ L^2( Ω) : a ≤ ∫_ Ω u(t) ≤ b\}$. We derive equivalent a posteriori error estimates with lower and upper bounds for both the state and the control approximation, which are used as indicators in adaptive multi-meshes finite element scheme. The error estimates are then implemented and tested with promising numerical experiments.

The Gauge-Uzawa method [GUM], which is a projection type algorithm to solve the time depend Navier-Stokes equations, has been constructed in [14] and enhanced in [15, 17] to apply to more complicated problems. Even though GUM possesses many advantages theoretically and numerically, the studies on GUM have been limited on the first order backward Euler scheme except normal mode error estimate in [16]. The goal of this paper is to research the 2nd order GUM. Because the classical 2nd order GUM which is studied in [16] needs rather strong stability condition, we modify GUM to be unconditionally stable method using BDF2 time marching. The stabilized GUM is equivalent to the rotational form of pressure correction method and the errors are already estimated in [8] for the Stokes equations. In this paper, we will evaluate errors of the stabilized GUM for the Navier-Stokes equations. We also prove that the stabilized GUM is an unconditionally stable method for the Navier-Stokes equations. So we conclude that the rotational form of pressure correction method in [8] is also unconditionally stable scheme and that the accuracy results in [8] are valid for the Navier-Stokes equations.

In this paper we consider the multiscale computation of a Steklov eigenvalue problem with rapidly oscillating coefficients. The new contribution obtained in this paper is a superapproximation estimate for solving the homogenized Steklov eigenvalue problem and to present a multiscale numerical method. Numerical simulations are then carried out to validate the theoretical results reported in the present paper.

We consider the finite element method for time dependent MHD flow at small magnetic Reynolds number. We make a second (and common) simplification in the model by assuming the time scales of the electrical and magnetic components are such that the electrical field responds instantaneously to changes in the fluid motion. This report gives a comprehensive error analysis for both the semi-discrete and a fully-discrete approximation. Finally, the effectiveness of the method is illustrated in several numerical experiments.

The paper is dedicated to the construction of an analytic solution for the level set equation in $R^2$ with an initial condition constituted by two half-planes. Such a problem can be seen as an equivalent Riemann problem in the Hamilton-Jacobi equation context. We first rewrite the level set equation as a non-strictly hyperbolic problem and obtain a Riemann problem where the line sharing the initial discontinuity corresponds to the half-planes junction. Three different solutions corresponding to a shock, a rarefaction and a contact discontinuity are given in function of the two half-planes configuration and we derive the solution for the level set equation. The study provides theoretical examples to test the numerical methods approaching the solution of viscosity of the level set equation. We perform simulations to check the three situations using a classical numerical method on a structured grid.

A novel phase field model for Willmore flow is proposed based on a nested variational time discretization. Thereby, the mean curvature in the Willmore functional is replaced by an approximate speed of mean curvature motion, which is computed via a fully implicit variational model for time discrete mean curvature motion. The time discretization of Willmore flow is then performed in a nested fashion: in an outer variational approach a natural time discretization is setup for the actual Willmore flow, whereas for the involved mean curvature the above variational approximation is taken into account. Hence, in each time step a PDE-constrained optimization problem has to be solved in which the actual surface geometry as well as the geometry resulting from the implicit curvature motion time step are represented by phase field functions. The convergence behavior is experimentally validated and compared with rigorously proved convergence estimates for a simple linear model problem. Computational results in 2D and 3D underline the robustness of the new discretization, in particular for large time steps and in comparison with a semi-implicit convexity splitting scheme. Furthermore, the new model is applied as a minimization method for elastic functionals in image restoration.

In this paper, we consider the extrapolation method for second order elliptic problems on general meshes and derive a type of finite element expansion which is dependent of the triangulation. It allows to prove the effectiveness of the extrapolation on general meshes and also validates the extrapolation method can be applied on the automatically produced meshes of the general computing domains. Some numerical examples are given to illustrate the theoretical analysis.

In this paper, we provide a priori $L^2$ error estimates for the semi-discrete discontinuous Galerkin method [3] and the local discontinuous Galerkin method [22] for one- and two-dimensional nonlinear Hamilton-Jacobi equations with smooth solutions. With a special Gauss-Radau projection, the optimal error estimates on rectangular meshes are obtained.

A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and boundary conditions is considered. The diffusion term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters.

The mechanochemical model proposed in 1983 by G. Oster, J. D. Murray and A. K. Harris has been deployed to describe various morphological phenomena in biology, such as feather bud formation [17] and angiogenesis and vasculogenesis [10, 13]. In this article, we apply a mechanochemical model to the formation of a somite to better understand the role that the mechanical aspects of the cells and the extracellular matrix (ECM) play in somitogenesis. In particular, our focus lies in the effect of the contractile forces generated by the cells, which are exerted onto the surrounding ECM. Our approach involves the linear stability analysis and a study of asymptotic behavior of the cell density based on a priori estimates. The full model considered in 2 dimensional space is numerically simulated to show that the traction force of the cells alone can generate a pattern.

The Kosloff & Kosloff (KK) absorbing-boundary method is shown to be a particular case of the split-PML method introduced by Bérenger. In its original form, the PML technique has been implemented for Maxwell's electromagnetic equations. On the other hand, the KK method was applied to the Schrödinger and acoustic wave equations. Both techniques have subsequently widely been used in dynamic elasticity, involving different rheological equations, including poroelasticity, and electromagnetism. The coordinate stretching used in the PML method is equivalent to the damping kernel in the KK method, which is based on the Maxwell viscoelastic model. Inside the absorbing strips, the result is a traveling wave which gradually attenuates without changing shape or undergoing dispersion. Moreover, we also show that the recently developed unsplit CPML method is based on the memory-variable formalism to describe anelasticity introduced by Carcione and co-workers, and that the damping kernel is based on the Zener viscoelastic model. The theoretical reflection coefficients, i.e., before discretization, are obtained and re-interpreted using the theory of viscoelasticity through the acoustic/electromagnetic analogy.

Staggered discontinuous Galerkin methods have been developed recently and are adopted successfully to many problems such as wave propagation, elliptic equation, convection-diffusion equation and the Maxwell's equations. For wave propagation, the method is proved to have the desirable properties of energy conservation, optimal order of convergence and block-diagonal mass matrices. In this paper, we perform an analysis for the dispersion error and the CFL constant. Our results show that the staggered method provides a smaller dispersion error compared with classical finite element method as well as non-staggered discontinuous Galerkin methods.