Virtual element method for quasilinear elliptic problems

Abstract

A virtual element method for the quasilinear equation $-\textrm {div} ({\boldsymbol \kappa }(u)\operatorname {grad} u)=f$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the $H^1$- and $L^2$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.

Optimal error estimates of coupled and divergence-free virtual element methods for the Poisson--Nernst--Planck/Navier--Stokes equations

“…Next, a duality argument for nonlinear elliptic equations [11], gives the following L 2 -error estimate…”

A Hybrid-High Order Method for Quasilinear Elliptic Problems of Nonmonotone Type

“…Finite element approximations for the above problem (4.1) is proposed and analyzed under various assumptions on the coefficient and data, and assumption on regularity for the solution, see [8-11, 40, 52, 60, 62]. Following [10,11,22,53], we make the following assumption on the solution of (4.1) as: Assumption Q.3. Assume the solution ∈ 1 0 (Ω) of (4.1) belongs to 2 (Ω) ∩ 1,∞ (Ω) for = 2 and belongs to 3 (Ω) for = 3.…”

“…For some related works on ℎ -DG method for strongly nonlinear elliptic problem, we refer [13,54] and references therein. We also refer articles on various nonlinear problem on second-order elliptic PDEs, Weak Galerkin methods of [63], Mimetic finite difference approximation of [2] and Virtual element method of [22]. The finite element approximations for the quasilinear problem are studied under various regularity assumptions on the coefficient and on the solution , for instance, in 2 and ∈ 2,2+ (Ω), > 0 or ∈ 2 (Ω), see [10,11,41,52,60,62,67].…”

A virtual element method for the von Kármán equations

“…Several virtual element methods based on conforming and non-conforming schemes have been developed to solve a wide variety of problems in Solid and Fluid Mechanics, for example [4][5][6]9,11,12,14,19,25,27,30,42,46,47]. Moreover, the VEM for thin structures has been developed in [16,24,29,30,44,45], whereas VEM for nonlinear problems have been introduced in [3,15,26,35,36,50] In this paper, we analyze a conforming 1 Virtual Element Method to approximate the isolated solutions of the von Kármán equations. We consider a variational formulation in terms of the transverse displacement and the Airy stress function, which contains bilinear and trilinear forms.…”