Weak Galerkin finite element method for a class of quasilinear elliptic problems

Solution of Exterior Quasilinear Problems Using Elliptical Arc Artificial Boundary

“…Combining with Lemma 1, we can deduce that A(u; u, v) + B(u; u, v) is also V-elliptic and bounded in V. By (3), a(x, u) is uniformly Lipschitz continuous with respect to u. Since these conditions hold, it is known [8] that variational problem (19) has a unique solution u ∈ V for all f ∈ L 2 (Ω). In the same way, we obtain that the variational problems ( 25) and ( 30) are uniquely solvable.…”

“…Problem (1) has numerous physical applications, e.g., in the field of magnetostatics, where u is the magnetic scalar potential and a is the magnetic permeability; in the field of compressible flow, where u is the velocity potential and a is the density. There have been many numerical results about problems of this kind in bounded domains, for example, the existence and uniqueness of weak solution [8,9], the finite element method [10][11][12], the mixed finite element method [13][14][15], the discontinuous Galerkin finite element method [16][17][18], the weak Galerkin finite element method [19], and the adaptive finite element method [20,21].…”

A weak Galerkin method and its two-grid algorithm for the quasi-linear elliptic problems of non-monotone type

“…Since then, there has been considerable interest in WG methods for the numerical solution of a wide range of partial differential equations. We refer the reader to [14,[16][17][18][19][20][21][22][23][24][25][26][27][30][31][32][33][34], and the references therein for details. However, there are few papers that are concerned to the nonlinear elliptic problems.…”

Stabilizer-free weak Galerkin methods for monotone quasilinear elliptic PDEs

“…However, the uniqueness and the error estimations of the numerical approximations are restricted only to the linear PDEs, and have not been addressed for the nonlinear ones. Only recently in [9], the authors gave the well-posedness and error estimate in the energy norm for the monotone quasilinear PDEs (1.1).…”