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 problems involving an energy concentrated on the vertices of an unknown polyhedral set, where the contours of the images to be recovered have preferred directions and focal points. We prove that such an energy is obtained as a Γ-limit of functionals defined on sets with smooth boundary that involve curvature terms of the boundary. The minimizers of the limit functional are polygons with edges either parallel to some prescribed directions or pointing to some fixed 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, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in 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 Laplacian of the liquid heights. We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analogues of these energy inequalities. Finally, we prove convergence of this approximation, and hence existence of a solution to this nonlinear degenerate parabolic system.

We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential 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 of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation uh on a partition of Ω of mesh size h = hL = 2-L satisfies the following bound in the streamline-diffusion norm $|||\cdot|||_{\rm SD}$, provided u belongs to the space $\mathcal{H}^{k+1}(\Omega)$ of functions with 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 conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion 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 a general 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 boundary element method with hypersingular operators on piecewise plane open or closed surfaces. The underlying meshes are supposed to be quasi-uniform. The solutions of problems on polyhedral or piecewise plane open surfaces exhibit typical singularities which limit the convergence rate of the boundary element method. On closed surfaces, and for sufficiently smooth given data, the solution is H1-regular whereas, on open surfaces, edge singularities are strong enough to prevent the solution from being in H1. In this paper we cover both cases and, in particular, prove an a priori error 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 of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to 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 the solid 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 least when 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 of topography source terms is obtained via a technique that includes these contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problems of 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 is the projection of the pre-flash potential along a piecewise harmonic eigenfunction which is constant along the lightning channel. The change in the potential outside the lightning channel is a harmonic function whose boundary conditions are expressed in terms of the pre-flash potential and the post-flash potential along the lightning channel. The expression for the lightning induced electric potential change is derived both for the continuous equations, and for a spatially discretized formulation of the continuous equations. The results for the continuous equations are based on the properties of the 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 $\langle \cdot, \cdot \rangle_{\Omega}$ is the inner product $ \langle \nabla u,\nabla v\rangle_\Omega =\int_{\Omega} \nabla u\cdot\nabla v \; {{\rm d}x}. $