Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems.
In: Applied Numerical Mathematics, Jg. 145 (2019-11-01), S. 361-383
academicJournal
Zugriff:
In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u ″ = f (x , u , u ′) , x ∈ [ a , b ] subject to some suitable boundary conditions at the endpoints x = a and x = b. We prove optimal L 2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p + 1 , when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p + 2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p , the (2 p + 1) th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]
Titel: |
Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems.
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Autor/in / Beteiligte Person: | Baccouch, Mahboub |
Zeitschrift: | Applied Numerical Mathematics, Jg. 145 (2019-11-01), S. 361-383 |
Veröffentlichung: | 2019 |
Medientyp: | academicJournal |
ISSN: | 0168-9274 (print) |
DOI: | 10.1016/j.apnum.2019.05.003 |
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