This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.

The motion of an incompressible fluid confined to a shallow basin witha slightly varying bottom topography is considered. Coriolis force,surface wind and pressure stresses, together with bottom andlateral friction stresses are taken into account. We introduceappropriate scalings into a three-dimensional anisotropic eddyviscosity model; after averaging on the vertical direction andconsidering some asymptotic assumptions, we obtain a two-dimensionalmodel, which approximates the three-dimensional model at the secondorder with respect to the ratio between the vertical scale and thelongitudinal scale. The derived model is shown to be symmetrizablethrough a suitable change of variables. Finally, we propose somenumerical tests with the aim to validate the proposed model.

The numerical modeling of the fully developed Poiseuille flowof a Newtonian fluid in a square section withslip yield boundary condition at the wall is presented.The stick regions in outer corners and the slip region in the center of the pipe faces are exhibited.Numerical computations cover the complete range of the dimensionless number describingthe slip yield effect, from a full slip to a full stick flow regime.The resolution of variational inequalitiesdescribing the flow is based on the augmented Lagrangian method and afinite element method. The localization of thestick-slip transition points is approximated by ananisotropic auto-adaptive mesh procedure.The singular behavior of the solution at the neighborhood of thestick-slip transition point is investigated.

A high-order compact finite difference scheme for a fully nonlinearparabolic differential equation is analyzed. The equation arises in themodeling of option prices in financial markets with transaction costs.It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation.The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

First–order accurate monotone conservative schemes have goodconvergence and stability properties, and thus play a veryimportant role in designing modern high resolution shock-capturingschemes.Do the monotone difference approximations alwaysgive a good numerical solution in sense of monotonicity preservationor suppression of oscillations? This note will investigate this problemfrom a numerical point of view and show thata (2K+1)-point monotone scheme may give an oscillatory solutioneven though the approximate solution is total variation diminishing, andsatisfies maximum principle as well as discrete entropy inequality.

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size.

This article presents the derivation of a semi-classical model of electromagnetic-wave propagation in a non centro-symmetric crystal. It consists of Maxwell's equations for the wave field coupled with a version of Bloch's equations which takes fully into account the discrete symmetry group of the crystal. The model is specialized in the case of a KDP crystal for which information about the dipolar moments at the Bloch level can be recovered from the macroscopic dispersion properties of the material.

This paper studies the gradient flow of a regularized Mumford-Shah functionalproposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L2 x L∞ initial data possesses a global weak solution, and it has a unique global in timestrong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H1 x H1 ∩ L∞ . A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence)of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates, which depend on $\frac{1}{{\varepsilon}}$ and $\frac{1}{k_{\varepsilon}}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^{\frac12})$ . Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\rm e}^{i \hat{s} \cdot \vec{v}}$ in terms of spherical harmonics $\{ Y_{\ell, m}(\hat{s}) \}_{|m|\le \ell\le \infty}$ . We consider the truncated series where the summation is performed over the $(\ell,m)$ 's satisfying $|m| \le \ell \le L$ . We prove that if $v = |\vec{v}|$ is large enough, the truncated series gives rise to an error lower than ϵ as soon as L satisfies $L+\frac{1}{2} \simeq v + CW^{\frac{2}{3}}(K \epsilon^{-\delta} v^\gamma )\, v^{\frac{1}{3}}$ where W is the Lambert function and $C\,, K, \,\delta, \, \gamma$ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation.

A general setting is proposed for the mixed finite element approximations ofelliptic differential problems involving a unilateral boundary condition. Thetreatment covers the Signorini problem as well as the unilateral contactproblem with or without friction. Existence, uniqueness for both thecontinuous and the discrete problem as well as error estimates are establishedin a general framework. As an application, the approximation of the Signoriniproblem by the lowest order mixed finite element method of Raviart–Thomas isproved to converge with a quasi-optimal error bound.

Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictatesthe interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε-1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.

We present an integral equation method for solving boundary valueproblems of the Helmholtz equation in unbounded domains. Themethod relies on the factorisation of one of the Calderón projectors by an operator approximating the exterioradmittance (Dirichlet to Neumann) operator of the scatteringobstacle. We show how the pseudo-differential calculus allows usto construct such approximations and that this yields integralequations without internal resonances and being well-conditionedat all frequencies. An implementation technique is elaborated,where again reasonings from pseudo-differential calculus play animportant rôle. Some numerical examples are presented which appearto confirm that the new integral equation leads to linear systemswhich are much better conditioned than the classical ("direct")integral equations and hence have much better behaviour whensolved with iterative techniques and matrix sparsification.

Arbitrage-free prices u of European contracts on risky assets whoselog-returns are modelled by Lévy processes satisfya parabolic partial integro-differential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$ .This PIDE is localized tobounded domains and the error due to this localization isestimated. The localized PIDE is discretized by theθ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for ${\mathcal{A}}$ can be replaced by a sparse matrix in the wavelet basis, and the linear systemsin each implicit time step are solved approximativelywith GMRES in linear complexity. The total work of the algorithm for M time steps is bounded byO(MN(log(N))2) operations and O(Nlog(N)) memory.The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solutionin the same complexity as finite difference approximationsof the standard Black–Scholes equation.Computational examples for various Lévy price processes are presented.

A Volterra model with mutual interferenceconcerning integrated pest management is proposed and analyzed. Byusing Floquet theorem and small amplitude perturbation method andcomparison theorem, we show the existence of a globallyasymptotically stable pest-eradication periodic solution. Further,we prove that when the stability of pest-eradication periodicsolution is lost, the system is permanent and there exists alocally stable positive periodic solution which arises from thepest-eradication periodic solution by bifurcation theory. When theunique positive periodic solution loses its stability, numericalsimulation shows there is a characteristic sequence ofbifurcations, leading to a chaotic dynamics. Finally, we comparethe validity of integrated pest management (IPM) strategy withclassical methods and conclude IPM strategy is more effective thanclassical methods.

Domain decomposition techniques provide a flexible tool for the numericalapproximation of partial differential equations. Here, we considermortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces.In particular, we focus on Lagrange multiplier spaceswhich yield optimal discretizationschemes and a locally supported basis for the associatedconstrained mortar spaces in case of hexahedral triangulations. As a result,standard efficient iterative solvers as multigrid methodscan be easily adapted to the nonconforming situation.We present the discretization errors in different norms for linear and quadratic mortar finite elements withdifferent Lagrange multiplier spaces.Numerical results illustrate the performance of our approach.

In this paper we analyze the effect of introducing a numerical integration in the piecewise linear finite element approximation of the Steklov eigenvalue problem. We obtain optimal order error estimates for the eigenfunctions when this numerical integration is used and we prove that, for singular eigenfunctions, the eigenvalues obtained using this reduced integration are better approximations than those obtained using exact integration when the mesh size is small enough.

We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega\subset \mathbb{R}^d$ and for T > 0 indimension dd ≥ 1. We use a wavelet based sparse gridspace discretization with mesh-width h and order pd ≥ 1, andhp discontinuous Galerkin time-discretization of order $r =O(\left|\log h\right|)$ on a geometric sequence of $O(\left|\log h\right|)$ many timesteps. The linear systems in each time step are solved iterativelyby $O(\left|\log h\right|)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2(Ω)-error of O(N-p) for u(x,T) where N is the total number of operations,provided that the initial data satisfies $u_0 \in H^\varepsilon(\Omega)$ with ε > 0 and that u(x,t) is smooth in x for t>0. Numerical experiments in dimension d up to 25 confirm thetheory.

The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version of the Picard iteration method. The accuracy ofthe proposed algorithm is investigated.

We consider a non-conforming stabilized domaindecomposition technique forthe discretization of the three-dimensional Laplace equation.The aim is to extend the numerical analysis of residual error indicators tothis model problem. Two formulations of the problem are consideredand the error estimators are studied for both. In thefirst one, the error estimator provides upper and lower bounds forthe energy norm of the mortar finite element solution whereas inthe second case, it also estimates the error for the Lagrangemultiplier.