scholarly journals Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form

2017 ◽  
Vol 37 (4) ◽  
pp. 4023-4054 ◽  
Author(s):  
Robert Eymard ◽  
Cindy Guichard
Keyword(s):  
Discontinuous Galerkin ◽  
Differential Operators ◽  
Divergence Form ◽  
Second Order

scholarly journals The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Alberto Cialdea ◽  
Vita Leonessa ◽  
Angelica Malaspina
Keyword(s):  
Dirichlet Problem ◽  
Differential Operators ◽  
Differential Forms ◽  
Variable Coefficients ◽  
Divergence Form ◽  
Boundary Integral ◽  
Second Order ◽  
Boundary Integral Method ◽  
Layer Potential ◽  
Partial Differential Operators

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

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.

Commutators with fractional differentiation for second-order elliptic operators on ℝn

2019 ◽  
Vol 22 (02) ◽  
pp. 1950010
Author(s):  
Yanping Chen ◽  
Qingquan Deng ◽  
Yong Ding
Keyword(s):  
Differential Operators ◽  
Divergence Form ◽  
Second Order ◽  
Bounded Mean Oscillation ◽  
Fractional Differentiation ◽  
Mean Oscillation ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

Let [Formula: see text] be a second-order divergence form elliptic operator and [Formula: see text] an accretive, [Formula: see text] matrix with bounded measurable complex coefficients in [Formula: see text] In this paper, we establish [Formula: see text] theory for the commutators generated by the fractional differential operators related to [Formula: see text] and bounded mean oscillation (BMO)–Sobolev functions.

DIV–CURL TYPE THEOREM, H-CONVERGENCE AND STOKES FORMULA IN THE HEISENBERG GROUP

2006 ◽  
Vol 08 (01) ◽  
pp. 67-99 ◽  
Author(s):  
BRUNO FRANCHI ◽  
NICOLETTA TCHOU ◽  
MARIA CARLA TESI
Keyword(s):  
Heisenberg Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Type Theorem ◽  
Divergence Form ◽  
Second Order ◽  
Order Differential Operator ◽  
Contact Manifolds ◽  
Stokes Formula ◽  
Horizontal Vector

In this paper, we prove a div–curl type theorem in the Heisenberg group ℍ1, and then we develop a theory of H-convergence for second order differential operators in divergence form in ℍ1. The div–curl theorem requires an intrinsic notion of the curl operator in ℍ1 (that we denote by curlℍ), that turns out to be a second order differential operator in the left invariant horizontal vector fields. As an evidence of the coherence of this definition, we prove an intrinsic Stokes formula for curlℍ. Eventually, we show that this notion is related to one of the exterior differentials in Rumin's complex on contact manifolds.

Sign in / Sign up

Export Citation Format

Share Document