Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations

Yadav, Sangita; Pani, Amiya K.; Park, Eun Jae

doi:10.1090/s0025-5718-2013-02662-2

Cited by 16 publications

(17 citation statements)

References 25 publications

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“…Moreover, it can be shown that j j j(x x x; y y y, ∇y y y), j j j M M M (x x x; y y y, ∇y y y), j j j M M MM M M (x x x; y y y, ∇y y y), j j j ∇M M M (x x x; y y y), j j j M M M∇M M M (x x x; y y y) are bounded for x x x ∈Ω, y y y ∈ O σ (I h M M M), by the same reasoning as in [17] , and let δM M M h ∈ X k , then we have the bound…”

Section: Lemma 44 (Energy Bound) Let M M Mmentioning

confidence: 67%

“…The demonstration of the stability follows closely the approach developed by [8,14,17,18] for linear and nonlinear elliptic problems. Since the problem is herein coupled, and as the elliptic operator is different, we report the modified main steps of the demonstrations that were initially developed in [17,18] for d = 2.…”

Section: Stability Of the Dg Formulationmentioning

confidence: 87%

“…Since the problem is herein coupled, and as the elliptic operator is different, we report the modified main steps of the demonstrations that were initially developed in [17,18] for d = 2. The main idea to prove the solution uniqueness and to establish the prior error estimates is to linearize the problem by reformulating the nonlinear problem in a fixed point form which is the solution of the linearized problem as proposed in [17,18,23].…”

Section: Stability Of the Dg Formulationmentioning

confidence: 99%

“…Since j j j is a twice continuously differential function with all the derivatives through the second order, which can be shown to be locally bounded in a ball around M M M ∈ R × R + 0 following the argumentation of [17,18] for d = 2, we denote by C y…”

Section: Stability Of the Dg Formulationmentioning

confidence: 99%

“…Moreover, he has shown that the solution of the adjoint problem suffers from sub-optimal rates of convergence when a NIPG formulation is used. Yadav et al [17] have extended the DG methods from a linear self-adjoint elliptic problem to a second order nonlinear elliptic problem. The nonlinear system resulting from DG methods is then analyzed based on a fixed point argument.…”

Section: Introductionmentioning

confidence: 99%

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A coupled electro-thermal Discontinuous Galerkin method

Homsi

Geuzaine

Noels

2017

Journal of Computational Physics

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This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems.In this paper we discuss the fundamental equations for the transport of electricity and heat, in terms of macroscopic variables such as temperature and electric potential. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients. The weak form can thus be formulated as a Discontinuous Galerkin method.The existence and uniqueness of the weak form solution are proved. The numerical properties of the nonlinear elliptic problems i.e., consistency and stability, are demonstrated under specific conditions, i.e. use of high enough stabilization parameter and at least quadratic polynomial approximations. Moreover the prior error estimates in the H 1 -norm and in the L 2 -norm are shown to be optimal in the mesh size with the polynomial approximation degree.

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Section: Lemma 44 (Energy Bound) Let M M Mmentioning

confidence: 67%

Section: Stability Of the Dg Formulationmentioning

confidence: 87%

Section: Stability Of the Dg Formulationmentioning

confidence: 99%

Section: Stability Of the Dg Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A coupled electro-thermal Discontinuous Galerkin method

Homsi

Geuzaine

Noels

2017

Journal of Computational Physics

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A Hybrid High‐Order method for finite elastoplastic deformations within a logarithmic strain framework

Abbas

Ern

Pignet

2019

Numerical Meth Engineering

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Summary We devise and evaluate numerically a Hybrid High‐Order (HHO) method for finite plasticity within a logarithmic strain framework. The HHO method uses as discrete unknowns piecewise polynomials of order k ≥ 1 on the mesh skeleton, together with cell‐based polynomials that can be eliminated locally by static condensation. The HHO method leads to a primal formulation, supports polyhedral meshes with nonmatching interfaces, and is free of volumetric locking. In addition, the integration of the behavior law is performed only at cell‐based quadrature nodes, and the tangent matrix in Newton's method is symmetric. Moreover, the principle of virtual work is satisfied locally with equilibrated tractions. Various two‐ and three‐dimensional benchmarks are presented, as well as comparisons against known solutions obtained with an industrial software using conforming and mixed finite elements.

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Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

Baccouch

2020

Numerical Methods Partial

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In this paper, we study the local discontinuous Galerkin (LDG) methods for two-dimensional nonlinear second-order elliptic problems of the type u xx + u yy = f (x, y, u, u x , u y), in a rectangular region Ω with classical boundary conditions on the boundary of Ω. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L 2 norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss-Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1, and for both mixed Dirichlet-Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal. KEYWORDS a priori error estimates, local discontinuous Galerkin method, nonlinear second-order elliptic boundary-value problems, supercloseness Dedicated to Professor Slimane Adjerid on the occasion of his 65th birthday.

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Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations

Cited by 16 publications

References 25 publications

A coupled electro-thermal Discontinuous Galerkin method

A coupled electro-thermal Discontinuous Galerkin method

A Hybrid High‐Order method for finite elastoplastic deformations within a logarithmic strain framework

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

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