hp-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems

Abstract

In this paper, we have analyzed a one parameter family of hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems −∇ · a(u, ∇u) + f (u, ∇u) = 0 with Dirichlet boundary conditions. These methods depend on the values of the parameter θ ∈ [−1, 1], where θ = +1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when a(u, ∇u) = ∇u and f (u, ∇u) = − f , that is, for the Poisson problem. The error estimate in the broken H 1 norm, which is optimal in h (mesh size) and suboptimal in p (degree of approximation) is derived using piecewise polynomials of degree p ≥ 2, when the solution u ∈ H 5/2 (Ω). In the case of linear elliptic problems also, this estimate is optimal in h and suboptimal in p. Further, optimal error estimate in the L 2 norm when θ = −1 is derived. Numerical experiments are presented to illustrate the theoretical results. Mathematics Subject Classification (2000) 65N12 · 65N30 · 65N15Supported by DST-DAAD (PPP-05) project.

Two‐Grid hp‐Version DGFEMs for Strongly Monotone Second‐Order Quasilinear Elliptic PDEs

“…Existence and uniqueness of the solution u H,P for this formulation is demonstrated in [12]. The formulation (10) is a symmetric interior penalty discretization of a linear elliptic PDE, where the coefficient µ(|∇ h u H,P |) is a known function; thereby, provided that the constant γ is chosen sufficiently large, the existence and uniqueness of the solution u 2G to this problem follows immediately, cf., for example, [14].…”

“…[12,13], for example. Remark 1.1 The SIP DGFEM scheme defined in (8) is identical to the method studied in [12], and represents a slight alternative to the parameterized DGFEMs considered in [13].…”

“…P r o o f. See [12] or [13]; we note, however, that the latter article employs a slightly different DGFEM formulation.…”

Numerical Properties of a Discontinuous Galerkin Fomulation for Electro-Thermal Coupled Problems

“…We have recently extended the DG method to solve electro-thermal coupled problems in terms of energetically conjugated fields gradients and fluxes [1], which to the authors knowledge, has not been introduced yet. In [1], the numerical properties of the nonlinear elliptic problem, following the method proposed in [2,5], have been derived. In particular, the convergence rates of the error in both the energy and L 2 -norms have been shown to be optimal with respect to the mesh size in terms of the polynomial degree approximation k (respectively in order k − 1 and k).…”

Anhp-Discontinuous Galerkin Method for the Optimal Control Problem of Laser Surface Hardening of Steel

“…For a review of work on DG methods for elliptic problems, we refer to [3,31]. [12][13][14] discuss DG methods for quasilinear and strongly non-linear elliptic problems. In [33,34], a non-symmetric interior penalty DGFEM is analyzed for elliptic and non-linear parabolic problems, respectively.…”