This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory).

We propose a model for segmentation problemsinvolving an energy concentrated on the vertices of an unknownpolyhedral set, where the contours of the images to be recoveredhave preferred directions and focal points.We prove that such an energy is obtained as a Γ-limit offunctionals defined on sets with smooth boundary thatinvolve curvature terms of the boundary.The minimizers of the limit functional are polygons withedges either parallel to some prescribed directions or pointing to somefixed points, that can also be taken as unknown of the problem.

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, ona solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants onboth the free liquid-liquid and liquid-air interfaces,and the presence of both attractive and repulsive van der Waals forcesin terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure,and a second energy inequality controlling the Laplacianof the liquid heights.We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analoguesof these energy inequalities. Finally, we prove convergence of this approximation,and hence existence of a solutionto this nonlinear degenerate parabolic system.

We develop the analysis of stabilized sparse tensor-productfinite element methods for high-dimensional,non-self-adjoint and possibly degenerate second-order partialdifferential equations of the form $-a:\nabla\nabla u + b \cdot \nabla u + cu = f(x)$ , $x \in\Omega = (0,1)^d \subset \mathbb{R}^d$ ,where $a \in \mathbb{R}^{d\times d}$ is a symmetric positive semidefinite matrix,using piecewise polynomials ofdegree p ≥ 1. Our convergence analysis is based on newhigh-dimensional approximation results in sparse tensor-productspaces. We show that the error between the analytical solution u and its stabilizedsparse finite element approximation u h on a partition ofΩ of mesh size h = hL = 2-L satisfies thefollowing bound in the streamline-diffusion norm $|||\cdot|||_{\rm SD}$ ,provided u belongs to the space $\mathcal{H}^{k+1}(\Omega)$ of functionswith square-integrable mixed (k+1)st derivatives: \[|||u-u_h|||_{\rm SD}\leq C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\} (|\sqrt{a}| h_L^t+ |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}} + c^{\frac{1}{2}} h_L^{t+1} \!)|u|_{\mathcal{H}^{t+1}(\Omega)}, \qquad \qquad \qquad\] where $\kappa_i=\kappa_i(p,t,L)$ , i=0,1, and $1 \leq t \leq \min(k,p)$ .We show, under various mild conditionsrelating L to p, L to d, or p to d,that in the case of elliptic transport-dominateddiffusion problems $\kappa_0, \kappa_1 \in (0,1)$ , and hence for p ≥ 1 the'error constant' $C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\}$ exhibits exponential decay as d → ∞; in the case of ageneral symmetric positive semidefinite matrix a,the error constant is shown to grow no faster than $\mathcal{O}(d^2)$ .In any case, in the absence of assumptions that relate L, p and d,the error $|||u - u_h|||_{\rm SD}$ is still bounded by $\kappa_\ast^{d-1}|\log_2 h_L|^{d-1}\mathcal{O}(|\sqrt{a}| h_L^t+ |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}}+c^{\frac{1}{2}} h_L^{t+1})$ , where $\kappa_\ast \in (0,1)$ for all L, p, d ≥ 2.

We prove an a priori error estimate for the hp-version of the boundaryelement method with hypersingular operators on piecewise plane open orclosed surfaces. The underlying meshes are supposed to be quasi-uniform.The solutions of problems on polyhedral or piecewise plane open surfaces exhibittypical singularities which limit the convergence rate of the boundary element method.On closed surfaces, and for sufficiently smooth given data, the solution isH 1-regular whereas, on open surfaces, edge singularities arestrong enough to prevent the solution from being in H 1.In this paper we cover both cases and, in particular, prove an a priorierror estimate for the h-version with quasi-uniform meshes.For open surfaces we prove a convergence like O(h1/2p-1),h being the mesh size and p denoting the polynomial degree.This result had been conjectured previously.

We study a depth-averaged model of gravity-driven flows made ofsolid grains and fluid, moving over variable basal surface. In particular, we are interested in applicationsto geophysical flows such as avalanches and debris flows,which typically contain both solid material and interstitial fluid.The model system consists of mass and momentum balance equations for thesolid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns,and by interphase drag source terms. The system is hyperbolic at leastwhen the difference between solid and fluid velocities is sufficiently small.We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Well-balancing oftopography source terms is obtained via a technique that includesthese contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problemsof perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.

The change in the electric potential due to lightning is evaluated.The potential along the lightning channel is a constant which isthe projection of the pre-flash potential along a piecewise harmoniceigenfunction which is constant along the lightning channel.The change in the potential outside the lightning channel is a harmonicfunction whose boundary conditionsare expressed in terms of the pre-flash potential andthe post-flash potential along the lightning channel.The expression for the lightning induced electric potential change isderived both for the continuous equations, and for a spatially discretizedformulation of the continuous equations.The results for the continuous equations are based on the properties ofthe eigenvalues and eigenfunctions of the following generalized eigenproblem:Find $u \in H_0^1 (\Omega)$ , $u \ne 0$ ,and $\lambda \in \mathbb{R}$ such that $\langle \nabla u, \nabla v \rangle_{\mathcal{L}} =\lambda \langle \nabla u, \nabla v \rangle_{\Omega}$ for all $v \in H_0^1 (\Omega)$ , where $\Omega \subset \mathbb{R}^n$ is a bounded domain (a box containing the thunderstorm), $\mathcal{L}$ is a subdomain (the lightning channel),and $\langle \cdot, \cdot \rangle_{\Omega}$ isthe inner product $\langle \nabla u,\nabla v\rangle_\Omega =\int_{\Omega}\nabla u\cdot\nabla v \; {{\rm d}x}.$