Let Ω ⊂ ℝN be a bounded domain and δ(x) be the distance of a point x ∈ Ω to the boundary. We study the positive solutions of the problem Δu + (μ/(δ(x)2))u = up in Ω, where p > 0, p ≠ 1 and μ ∈ ℝ, μ ≠ 0 is smaller than the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper, we first give the complete picture of the radial solutions in balls. In particular, we establish for p > 1 the existence of a unique large solution behaving like δ−(2/(p−1)) at the boundary. In general domains, we extend the results of Bandle and Pozio and show that there exists a unique singular solutions u such that $u/\delta ^{\beta _-}\to c$ on the boundary for an arbitrary positive function $c \in C^{2+\gamma }(\partial \Omega ) \, (\gamma \in (0,1)), c \ges 0$. Here β− is the smaller root of β(β − 1) + μ = 0.

This paper proposes an extrapolation cascadic multigrid (EXCMG) method to solve elliptic problems in domains with reentrant corners. On a class of λ-graded meshes, we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids (current and previous grids). Then, this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient method. Recursive application of this idea results in the EXCMG method proposed in this paper. Finally, numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.

In this article we develop a posteriori error estimates for second orderlinear elliptic problems with point sources in two- and three-dimensional domains. Weprove a global upper bound and a local lower bound for the error measured in a weightedSobolev space. The weight considered is a (positive) power of the distance to the supportof the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theoryhinges on local approximation properties of either Clément or Scott–Zhang interpolationoperators, without need of modifications, and makes use of weighted estimates forfractional integrals and maximal functions. Numerical experiments with an adaptivealgorithm yield optimal meshes and very good effectivity indices.

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal L2-error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.

Let Ω⊂ℝN be a smooth bounded domain and let f⁄≡0 be a possibly discontinuous and unbounded function. We give a necessary and sufficient condition on f for the existence of positive solutions for all λ>0 of Dirichlet periodic parabolic problems of the form Lu=h(x,t,u)+λf(x,t), where h is a nonnegative Carathéodory function that is sublinear at infinity. When this condition is not fulfilled, under some additional assumptions on h we characterize the set of λs for which the aforementioned problem possesses some positive solution. All results remain true for the corresponding elliptic problems.

We obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger-type operators. One of the main results of the paper concerns solvability conditions for the equation –Δu + V(x)u–au = f where a ≥ 0. The conditions are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

Using variational methods, we establish the existence and multiplicity of positive solutions for the following class of problems:

where λ,β∈(0,∞), q∈(1,2*−1), 2*=2N/(N−2), N≥3, V,Z:ℝN→ℝ are continuous functions verifying some hypotheses.

We present new a posteriori error estimates for the finite volume approximationsof elliptic problems. They are obtained by applying functional a posteriorierror estimates to natural extensions of the approximate solution and its fluxcomputed by the finite volume method. The estimates give guaranteed upper boundsfor the errors in terms of the primal (energy) norm, dual norm (for fluxes), andalso in terms of the combined primal-dual norms. It is shown that the estimatesprovide sharp upper and lower bounds of the error and their practicalcomputation requires solving only finite-dimensional problems.

In this paper, we present numerical methodsfor the determination of solitons, that consist in spatially localizedstationary states of nonlinear scalar equations or coupled systemsarising in nonlinear optics.We first use the well-known shooting method in order to findexcited states (characterized by the number k of nodes) for theclassical nonlinear Schrödinger equation. Asymptotics can thenbe derived in the limits of either large k are large nonlinear exponents σ.In a second part, we compute solitons for a nonlinearsystem governing the propagation of two coupled waves in a quadratic media in anyspatial dimension, starting from one-dimensional states obtained with a shooting method and considering the dimension as a continuation parameter. Finally, we investigate the case of three wavemixing, for which the shooting method is not relevant.