An hp-version of the discontinuous Galerkin time-stepping method for Volterra integral equations with weakly singular kernels
| Autor: | Lijun Yi, Hongjiong Tian, Lina Wang |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Numerical Analysis
Applied Mathematics Volterra integral equation Exponential function Computational Mathematics symbols.namesake Discontinuous Galerkin method Norm (mathematics) symbols Applied mathematics A priori and a posteriori Polygon mesh Gravitational singularity Algebraic number Mathematics |
| Zdroj: | Applied Numerical Mathematics. 161:218-232 |
| ISSN: | 0168-9274 |
| DOI: | 10.1016/j.apnum.2020.11.006 |
| Popis: | We develop and analyze an hp-version of the discontinuous Galerkin time-stepping method for linear Volterra integral equations with weakly singular kernels. We derive a priori error bound in the L 2 -norm that is fully explicit in the local time steps, the local approximation orders, and the local regularity of the exact solutions. For solutions with singular behavior near t = 0 caused by the weakly singular kernels, we prove optimal algebraic convergence rates for the h-version of the discontinuous Galerkin approximations on graded meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometrically refined time steps and linearly increasing approximation orders. Numerical experiments are presented to illustrate the theoretical results. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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