Abstract
AbstractIn this article we consider the a posteriori error analysis of hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of strongly monotone type. In particular, we employ and analyze a practical solution scheme based on exploiting a discrete Kačanov iterative linearization. The resulting a posteriori error bound explicitly takes into account the three sources of error: discretization, linearization, and linear solver errors. Numerical experiments are presented to demonstrate the practical performance of the proposed hp-adaptive refinement strategy.
Highlights
In this article, we consider the a posteriori error analysis, in a natural meshdependent energy norm, for a class of interior-penalty hp-version discontinuous Galerkin finite element methods (DGFEMs) for the numerical solution of the following quasilinear elliptic boundary value problem:−∇ · (μ(x, |∇u|)∇u) = f in Ω, u = 0 on Γ. (1)Here, Ω ⊂ R2 is a bounded polygon with a Lipschitz continuous boundary Γ, and f ∈ L2(Ω), where for an open set D ⊆ Ω, we signify by L2(D) the space of all square integrable functions on D
We assume that the nonlinearity μ satisfies the following assumptions: (A1) μ ∈ C0(Ω × [0, ∞)); (A2) there exist positive constants mμ, Mμ such that mμ(t − s) ≤ μ(x, t)t − μ(x, s)s ≤ Mμ(t − s), t ≥ s ≥ 0, x ∈ Ω
We suppose that Th is regularly reducible, i.e., there exists a shape-regular conforming mesh Th such that the closure of each element in Th is a union of closures of elements of Th, and that there exists a constant C > 0, independent of the element sizes, such that for any two elements κ ∈ Th and κ ∈ Th with κ ⊆ κ we have hκ/hκ ≤ C
Summary
We consider the a posteriori error analysis, in a natural meshdependent energy norm, for a class of interior-penalty hp-version discontinuous Galerkin finite element methods (DGFEMs) for the numerical solution of the following quasilinear elliptic boundary value problem:. We allow Th to be 1-irregular, i.e., each edge of any one element κ ∈ Th contains at most one hanging node (which, for simplicity, we assume to be the midpoint of the corresponding edge) In this context, we suppose that Th is regularly reducible (cf [18, Section 7.1] and [12]), i.e., there exists a shape-regular conforming (regular) mesh Th (consisting of triangles and parallelograms) such that the closure of each element in Th is a union of closures of elements of Th, and that there exists a constant C > 0, independent of the element sizes, such that for any two elements κ ∈ Th and κ ∈ Th with κ ⊆ κ we have hκ/hκ ≤ C.
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