An $hp$-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type

Abstract

Abstract. In this paper, an hp-local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On hp-quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken H 1 norm and L 2 norm which are optimal in h, suboptimal in p are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results.

Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations

“…(5.19) Since [[φ]] = 0 on Γ I and φ = 0 on boundary of Ω, we rewrite the second term on the right-hand of (5. 19) as…”

“…It is again observed in [9] that if C 11 and C 22 are of order one, then u h and q h converge in L 2 -norm with order k + 1 and k + 1/2, respectively, for any k ≥ 0. Subsequently, the authors in [19] have discussed the LDG method for quasilinear elliptic boundary value problems and they have shown that u h and q h converge in L 2 -norm with order k + 1 and k, respectively. For the LDG method applied to nonlinear elliptic problems, we also refer to [7].…”

“…Finally, with help of a suitable postprocessing of the approximation to the potential, which results in solving a linear elliptic problem, a superconvergence result for the potential is derived. We note that intermediate projections facilitate the error analysis for nonlinear elliptic problems (see [17]- [19], [20]- [21] and for parabolic problems [23]), where an elliptic projection as an intermediate projection was first introduced.…”

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.…”

A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integro-differential equations

“…In [5], optimal order of convergence of LDG method applied to a Poisson equation has been established. Subsequently, Perugia and Schötzau [18] have discussed a priori hp-error estimates for linear elliptic problems and then, Gudi et al [12] have derived hp-error estimates for nonlinear elliptic problems. For higher order partial differential equations using LDG method, see [7,9,13,25,26] and references, therein.In this paper, hp-DG methods which, in particular, include the original LDG scheme, are applied to the problem (1.1)-(1.4).…”