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This article analyzes the effect of the penalty parameter used in symmetric dual-wind discontinuous Galerkin (DWDG) methods for approximating second order elliptic partial differential equations (PDE). DWDG methods follow from the DG differential calculus framework that defines discrete differential operators used to replace the continuous differential operators when discretizing a PDE. We establish the convergence of the DWDG approximation to a continuous Galerkin approximation as the penalty parameter tends towards infinity. We also test the influence of the regularity of the solution for elliptic second-order PDEs with regards to the relationship between the penalty parameter and the error for the DWDG approximation. Numerical experiments are provided to validate the theoretical results and to investigate the relationship between the penalty parameter and the L^2-error.For more information see https://ejde.math.txstate.edu/conf-proc/26/l1/abstr.html
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On the convergence of Fourier spectral methods involving noncompact operators
Abstract Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $$\mathcal L u = f$$. The framework posits the existence of a left-Fredholm regulator for $$\mathcal L$$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann–Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.
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- Award ID(s):
- 2306438
- PAR ID:
- 10525691
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- IMA Journal of Numerical Analysis
- Volume:
- 45
- Issue:
- 3
- ISSN:
- 0272-4979
- Format(s):
- Medium: X Size: p. 1743-1785
- Size(s):
- p. 1743-1785
- Sponsoring Org:
- National Science Foundation
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