A hierarchical preconditioner for the mortar finite element method.
A Hölder infinity Laplacian
In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.
A Hölder infinity Laplacian
In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.
A homogenization result for planar, polygonal networks
A homotopy method for solving an equation of the type
A la recherche du spectre perdu: An invitation to nonlinear spectral theory
We give a survey on spectra for various classes of nonlinear operators, with a particular emphasis on a comparison of their advantages and drawbacks. Here the most useful spectra are the asymptotic spectrum by M. Furi, M. Martelli and A. Vignoli (1978), the global spectrum by W. Feng (1997), and the local spectrum (called “phantom”) by P. Santucci and M. Väth (2000). In the last part we discuss these spectra for homogeneous operators (of any degree), and derive a discreteness result and a nonlinear...
A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two
We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the...
A Legendre spectral collocation method for the biharmonic Dirichlet problem
A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem
A Legendre spectral collocation method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplacian of the solution on the two vertical sides of the square. The Schur complement system is solved by a...
A limiting geometry for capillary surfaces
A Liouville theorem for nonlinear elliptic systems with isotropic nonlinearities
A Liouville theorem for solutions of the Monge–Ampère equation with periodic data
A Liouville-type theorem for elliptic systems
A Liouville-type theorem for the p-laplacian with potential term
A Liouville-type theorem for very weak solutions of nonlinear partial differential equations.
A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem
Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the norm of |∇u| is dominated by the norms of and . If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.
A localized orthogonal decomposition method for semi-linear elliptic problems
In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H) | where H is the coarse mesh size. Without any assumptions...
A lower bound for the gradient of -harmonic functions.
A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients
In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci. 342 (2006) 883–886; CALCOLO 45 (2008) 111–147; J. Sci. Comput. 38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditions to obtain fast convergence of domain decomposition...
A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients
In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci.342 (2006) 883–886; CALCOLO45 (2008) 111–147; J. Sci. Comput.38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditions to obtain fast convergence of domain decomposition...