scholarly journals Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1

2006 ◽  
Vol 105 (3) ◽  
pp. 337-374 ◽  
Author(s):  
J. Casado-Díaz ◽  
T. Chacón Rebollo ◽  
V. Girault ◽  
M. Gómez Mármol ◽  
F. Murat
Keyword(s):  
Finite Elements ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Second Order ◽  
Order Linear ◽  
Right Hand

scholarly journals Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1

2018 ◽  
Vol 2018 ◽  
pp. 1-15
Author(s):  
Abdeluaab Lidouh ◽  
Rachid Messaoudi
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Divergence Form ◽  
Second Order ◽  
Discrete Problem ◽  
Renormalized Solution ◽  
Order Linear ◽  
Right Hand ◽  
The Right

We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω and the right-hand side belongs to L1Ω; we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in W01,qΩ for every q with 1≤q<d/d-1 (d=2 or d=3) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate in W01,qΩ when the right-hand side f belongs to LrΩ verifying Tkf∈H1Ω for every k>0, for some r>1.

scholarly journals The Dirichlet Problem for elliptic equations in unbounded domains of the plane

2008 ◽  
Vol 6 (1) ◽  
pp. 47-58 ◽  
Author(s):  
Paola Cavaliere ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Existence Theorem ◽  
Elliptic Equations ◽  
Unbounded Domains ◽  
Suitable Condition ◽  
Second Order ◽  
Order Linear ◽  
Uniqueness And Existence

In this paper we prove a uniqueness and existence theorem for the Dirichlet problem inW2,pfor second order linear elliptic equations in unbounded domains of the plane. Here the leading coefficients are locally of classVMOand satisfy a suitable condition at infinity.

Perturbation and Solvability of Initial Lp Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains

2014 ◽  
Vol 66 (2) ◽  
pp. 429-452 ◽  
Author(s):  
Jorge Rivera-Noriega
Keyword(s):  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Carleson Measure ◽  
Divergence Form ◽  
Second Order ◽  
Linear Operators ◽  
Dirichlet Problems ◽  
Small Perturbations ◽  
Cylindrical Domains

AbstractFor parabolic linear operators L of second order in divergence form, we prove that the solvability of initial Lp Dirichlet problems for the whole range 1 < p < ∞ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of L satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of p > 1, the initial Lp Dirichlet problem associated with Lu = 0 over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

Sign in / Sign up

Export Citation Format

Share Document