Optimal error estimates of an IPDG scheme for the resistive magnetic induction equation

Tanmay Sarkar ()
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Tanmay Sarkar: Indian Institute of Technology Jammu

Partial Differential Equations and Applications, 2023, vol. 4, issue 4, 1-33

Abstract: Abstract In this paper, we develop the framework for error analysis of a fully-discrete interior penalty discontinuous Galerkin (IPDG) scheme designed for the initial-boundary value problem associated with the resistive magnetic induction equation. We demonstrate the error estimates for semi-discrete IPDG schemes, in which the obtained convergence rates are optimal in the energy norm, but sub-optimal in the $$L^2$$ L 2 -norm. For sufficiently smooth solution, we derive optimal a-priori error estimates in the $$L^2$$ L 2 -norm $$\mathcal {O}(h^{1+l})$$ O ( h 1 + l ) , where l denotes the polynomial degree and h mesh size. Furthermore, we extend the error analysis to the fully-discrete schemes. For the fully-discrete schemes, the optimal convergence rates are obtained in the energy norm and $$L^2$$ L 2 -norm for both space and time using the backward Euler and second order backward difference schemes for time discretization.

Keywords: Discontinuous Galerkin methods; Magnetic induction; Resistivity; Backward difference formula; Error analysis; Rate of convergence; 65M60; 65M12; 65M15; 76W05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00245-z

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