We consider the numerical approximation of a first order stationary hyperbolic equation by the method of characteristics with pseudo time step using discontinuous finite elements on a mesh ${𝒯}_h$. For this method, we exhibit a “natural” norm || || for which we show that the discrete variational problem $P_h^k$ is well posed and we obtain an error estimate. We show that when goes to zero problem $\left(P_h^k\right)$ (resp. the || || norm) has as a limit problem ( ) (resp. the || || norm) associated...

Let be an odd function of a class C such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f\left(x\right)/x$ increases on . We approximate the positive solution of Δ = 0, on ${}_{+}^2$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_L+f\left(u_L\right)=0,$ on ]0,[ with adequate non-homogeneous Dirichlet conditions. We show that the error tends to zero exponentially fast, in the uniform norm.

Let be a distribution function (d.f) in the domain of attraction of an extreme value distribution $H_{\gamma }$; it is well-known that , where is the d.f of the excesses over , converges, when tends to , the end-point of , to $G_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where $G_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function which verifies ${lim}_{u\to s_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}(x/\sigma \left(u\right))|$ converges to faster than $d\left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\gamma }(x/\sigma \left(u\right))|$.

In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand : × $\to$ $\cup$ $\left\{\infty \right\}$, where is the -dimensional Euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.