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A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

Author

Listed:
  • Brdar, Mirjana
  • Franz, Sebastian
  • Ludwig, Lars
  • Roos, Hans-Görg

Abstract

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

Suggested Citation

  • Brdar, Mirjana & Franz, Sebastian & Ludwig, Lars & Roos, Hans-Görg, 2023. "A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space," Applied Mathematics and Computation, Elsevier, vol. 437(C).
    • Handle: RePEc:eee:apmaco:v:437:y:2023:i:c:s0096300322005811
    DOI: 10.1016/j.amc.2022.127507
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    References listed on IDEAS

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    1. V. Y. Glizer, 2000. "Asymptotic Solution of a Boundary-Value Problem for Linear Singularly-Perturbed Functional Differential Equations Arising in Optimal Control Theory," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 309-335, August.
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