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Numerical Methods for Elliptic PDEs
Published in Victor G. Ganzha, Evgenii V. Vorozhtsov, Numerical Solutions for Partial Differential Equations, 2017
Victor G. Ganzha, Evgenii V. Vorozhtsov
Let us illustrate the basic ideas of the finite element method at the example of the elliptic boundary-value problem () Δu=−f(x,y),(x,y)∈D () u|Γ=0 where () D=(a,b)×(c,d),−∞< a< b< ∞, −∞<c<d<∞
Iterative Methods for Solving Linear Systems
Published in Victor S. Ryaben’kii, Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, 2006
Victor S. Ryaben’kii, Semyon V. Tsynkov
obtained as a discrete approximation of an elliptic boundary value problem. For example, it may be a system of finite-difference equations introduced in Section 5.1.3. In doing so, the better the operator An approximates the original elliptic differential operator, the higher the dimension n of the space ℝn is. As such, we are effectively dealing with a sequence of approximating spaces ℝn,n→∞, that we will assume Euclidean with the scalar product (x,y)(n).
Elliptic Equations: Equilibrium in Two Dimensions
Published in Saad A. Ragab, Hassan E. Fayed, Introduction to Finite Element Analysis for Engineers, 2018
Saad A. Ragab, Hassan E. Fayed
and classify the boundary conditions as natural or essential. It is recommended that one multiplies the differential equation by a test function w(x,y) $ w(x, y) $ , integrate over the square, and proceed showing the details of the integration by parts and applications of boundary conditions.Derive the weak form for the elliptic boundary-value problem
REGINN-IT method with general convex penalty terms for nonlinear inverse problems
Published in Applicable Analysis, 2022
In this example, we consider the identification of space-dependent coefficient c in the elliptic boundary value problem from the measurement of u in Ω, where , , is a bounded domain with Lipschitz boundary, and . We assume is the sought solution. This problem can reduce to solving an nonlinear ill-posed problem of the form (1) with where is the unique solution of (48). It is well known that there exists a constant such that F is well-defined on It is easy to show that is Fréchet differentiable. Its Fréchet derivative and the adjoint of its derivative are given by where is defined by . Furthermore, Assumption 3.1(d) holds (see[33]).
Existence of non-zero solutions for a Dirichlet problem driven by (p(x),q(x)-Laplacian
Published in Applicable Analysis, 2022
A. Chinnì, A. Sciammetta, E. Tornatore
The aim of this paper is to study the following nonlinear elliptic boundary value problem with variable exponent where and are the -Laplacian operator and the -Laplacian operator, respectively, λ is a positive real parameter, is an open bounded domain with a Lipschitz boundary and p and satisfying the following condition The nonlinear term is a Carathéodory function, i.e. is measurable for every and is continuous for almost every .
Small diffusion and short-time asymptotics for Pucci operators
Published in Applicable Analysis, 2022
Diego Berti, Rolando Magnanini
Varadhan's formulas are now more than fifty years old. Their original motivation has to do with the asymptotic behavior of probabilities. The two important reference situations concern an elliptic boundary value problem, and a parabolic initial-boundary value problem, Here, Ω is a domain in , , not necessarily bounded and with sufficiently regular boundary Γ, and is an elliptic operator. In both cases, the maximum principle gives that the values of and v in Ω belong to the interval , thus giving grounds for a probabilistic interpretation for them.