Effective solution of the Carleman problem for some groups of divergent type.
Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods
We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a...
Efficient rectangular mixed finite elements in two and three space variables
Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations
In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function...
Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients
Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of...
Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps
Asymptotics with sharp remainder estimates are recovered for number of eigenvalues of the generalized Maxwell problem and for related Laplacians which are similar to Neumann Laplacian. We consider domains with ultra-thin cusps (with ) width ; ) and recover eigenvalue asymptotics with sharp remainder estimates.
Eigenvalue problems and bifurcation of nonhomogeneous semilinear elliptic equations in exterior strip domains.
Eigenvalue stability bounds via weighted Sobolov spaces.
Elasto-plastic torsion problem as an infinity Laplace's equation.
Element-oriented and edge-oriented local error estimators for nonconforming finite element methods
Elliptic Equations with Noninvertible Fredholm Linear Part and Bounded Nonlinearities.
Elliptic functions, area integrals and the exponential square class on B₁(0) ⊆ ℝⁿ, n > 2
For two strictly elliptic operators L₀ and L₁ on the unit ball in ℝⁿ, whose coefficients have a difference function that satisfies a Carleson-type condition, it is shown that a pointwise comparison concerning Lusin area integrals is valid. This result is used to prove that if L₁u₁ = 0 in B₁(0) and then lies in the exponential square class whenever L₀ is an operator so that L₀u₀ = 0 and implies is in the exponential square class; here S is the Lusin area integral. The exponential square theorem,...
Elliptic perturbations for Hammerstein equations with singular nonlinear term.
Elliptic problems in generalized Orlicz-Musielak spaces
We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a Δ2 nor ∇2-condition for an inhomogeneous and anisotropic N-function but assume it to be log-Hölder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ∞-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.
Elliptic problems with nonmonotone discontinuities at resonance.
Embedding and a priori wavelet-adaptivity for Dirichlet problems
Embedding and a priori wavelet-adaptivity for Dirichlet problems
The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problems suffers under a coarse boundary approximation. To overcome this drawback the method is furnished with an a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme. In contrast to similar,...
Embedding of open riemannian manifolds by harmonic functions
Let be a noncompact Riemannian manifold of dimension . Then there exists a proper embedding of into by harmonic functions on . It is easy to find harmonic functions which give an embedding. However, it is more difficult to achieve properness. The proof depends on the theorems of Lax-Malgrange and Aronszajn-Cordes in the theory of elliptic equations.
En ε-uniformly convergent non-conforming finite element method for a singularly perturbed elliptic problem in two dimensions
Enhanced electrical impedance tomography via the Mumford–Shah functional
We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is well-known that this problem is highly illposed. In this work, we propose the use of the Mumford–Shah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several...