TY - JOUR
T1 - Superconvergence of an Ultra-Weak Discontinuous Galerkin Method for Nonlinear Second-Order Initial-Value Problems
AU - Baccouch, Mahboub
N1 - Funding Information:
This research was supported by the NASA Nebraska Space Grant (Federal Grant/Award Number 80NSSC20M0112).
Publisher Copyright:
© 2023 World Scientific Publishing Company.
PY - 2023/3/1
Y1 - 2023/3/1
N2 - In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form u″ = f(x,u). Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order p + 1 in the L2-norm for the solution, when piecewise polynomials of degree at most p are used. We use these results to prove that the UWDG solution is superconvergent with order p + 2 for p = 2 and p + 3 for p ≥ 3 towards a special projection of the exact solution. We further prove that the p-degree UWDG solution and its derivative are O(h2p) superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree p ≥ 2. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.
AB - In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form u″ = f(x,u). Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order p + 1 in the L2-norm for the solution, when piecewise polynomials of degree at most p are used. We use these results to prove that the UWDG solution is superconvergent with order p + 2 for p = 2 and p + 3 for p ≥ 3 towards a special projection of the exact solution. We further prove that the p-degree UWDG solution and its derivative are O(h2p) superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree p ≥ 2. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.
KW - Second-order initial-value problems
KW - a priori error estimate
KW - superconvergence
KW - ultra-weak discontinuous Galerkin method
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U2 - 10.1142/S0219876222500426
DO - 10.1142/S0219876222500426
M3 - Article
AN - SCOPUS:85139864446
SN - 0219-8762
VL - 20
JO - International Journal of Computational Methods
JF - International Journal of Computational Methods
IS - 2
M1 - 2250042
ER -