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.
This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Kármán equations defined on a polygonal domain. A discrete inf-sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established and this allows the proof of local existence and uniqueness of a discrete solution to the non-linear problem with a Banach fixed point theorem. The Newton scheme is locally secondorder convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel C 0 interior penalty method (IPDG). In contrast to the known C 0 -IPDG due to Brenner et al [9], the overall discrete formulation maintains symmetry of the trilinear form in the first two components -despite the general non-symmetry of the discrete nonlinear problems. Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known C 0 -IPDG lead to complications with some non-residual type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a non convex domain.
The gradient discretisation method is a generic framework that is applicable to a number of schemes for diffusion equations, and provides in particular generic error estimates in L 2 and H 1 -like norms. In this paper, we establish an improved L 2 error estimate for gradient schemes. This estimate is applied to a family of gradient schemes, namely, the Hybrid Mimetic Mixed (HMM) schemes, and yields an O(h 2 ) super-convergence rate in L 2 norm, provided local compensations occur between the cell points used to define the scheme and the centers of mass of the cells. To establish this result, a modified HMM method is designed by just changing the quadrature of the source term; this modified HMM enjoys a super-convergence result even on meshes without local compensations. Finally, the link between HMM and Two-Point Flux Approximation (TPFA) finite volume schemes is exploited to partially answer a long-standing conjecture on the super-convergence of TPFA schemes.Proof. The proof hinges on two tricks. In Step 1, letting u D * be the solution to the modified HMM scheme, we show thatCombined with the result from Step 1 and the super-convergence property (4.8) of the modified HMM method, this concludes the proof.Step 1: Comparison of the solutions to the schemes for D and D * . Let u D * be the solution to (2.1) with D * instead of D. Subtracting the two gradient schemes corresponding to D * and D we see that, for all v D ∈ X D,0 , Ω
We consider a system of second-order nonlinear elliptic partial differential equations that models the equilibrium configurations of a two-dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin (dG) finite element methods are used to approximate the solutions of this nonlinear problem with nonhomogeneous Dirichlet boundary conditions. A discrete inf–sup condition demonstrates the stability of the dG discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the nonlinear problem. A priori error estimates in the energy and $\mathbf{L}^2$ norms are derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of the Newton iterates along with complementary numerical experiments.
The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of two-dimensional Navier–Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a priori error analysis and an a posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data, and so has to provide discrete stability by a perturbation analysis before the Newton–Kantorovich theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix–Raviart FEM for second-order linear nonselfadjoint and indefinite elliptic problems with $L^\infty $ coefficients. The paper identifies six parameters and sufficient conditions for the local a priori and a posteriori error control of conforming and nonconforming discretizations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.
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.
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