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𝐴 - 𝑃𝑂𝑆𝑇𝐸𝑅𝐼𝑂𝑅𝐼 error estimates for linear exterior problems 𝑉𝐼𝐴 mixed-FEM and DtN mappings

Mauricio A. Barrientos, Gabriel N. Gatica, Matthias Maischak (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the...

3D domain decomposition method coupling conforming and nonconforming finite elements

Abdellatif Agouzal, Laurence Lamoulie, Jean-Marie Thomas (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper deals with the solution of problems involving partial differential equations in  3 . For three dimensional case, methods are useful if they require neither domain boundary regularity nor regularity for the exact solution of the problem. A new domain decomposition method is therefore presented which uses low degree finite elements. The numerical approximation of the solution is easy, and optimal error bounds are obtained according to suitable norms.

A 3G-Theorem for Jordan Domains in ℝ²

Lotfi Riahi (2004)

Colloquium Mathematicae

We prove a new 3G-Theorem for the Laplace Green function G on an arbitrary Jordan domain D in ℝ². This theorem extends the recent one proved on a Dini-smooth Jordan domain.

A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri (2003)

Colloquium Mathematicae

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ( P λ ) ⎩ u Ω = 0 where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ( P λ ) admits a non-zero, non-negative strong solution u λ p 2 W 2 , p ( Ω ) such that l i m λ 0 | | u λ | | W 2 , p ( Ω ) = 0 for all p ≥ 2. Moreover, the function λ I λ ( u λ ) is negative and decreasing in ]0,λ*[, where I λ is the energy functional related to ( P λ ).

A boundary integral equation for Calderón's inverse conductivity problem.

Kari Astala, Lassi Päivärinta (2006)

Collectanea Mathematica

Towards a constructive method to determine an L∞-conductivity from the corresponding Dirichlet to Neumann operator, we establish a Fredholm integral equation of the second kind at the boundary of a two dimensional body. We show that this equation depends directly on the measured data and has always a unique solution. This way the geometric optics solutions for the L∞-conductivity problem can be determined in a stable manner at the boundary and outside of the body.

A brief review of some application driven fast algorithms for elliptic partial differential equations

Prabir Daripa (2012)

Open Mathematics

Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.

A C1-P2 finite element without nodal basis

Shangyou Zhang (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic...

A calculus for a class of finitely degenerate pseudodifferential operators

Ingo Witt (2003)

Banach Center Publications

For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate.

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