Stability of Higher-Order ALE-STDGM for Nonlinear Problems in Time-Dependent Domains

  • Monika BalázsováEmail author
  • Miloslav Vlasák
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


In this paper we investigate the stability of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear convection-diffusion problem in time-dependent domains. At first we define the continuous problem and reformulate it using the Arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so called ALE-derivative and an additional convective term. Then the problem is discretized with the aid of the ALE space-time discontinuous Galerkin method (ALE-STDGM). The discretization uses piecewise polynomial functions of degree p ≥ 1 in space and q > 1 in time. Finally in the last part of the paper we present our results concerning the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties, namely its continuity in the \(\Vert \cdot \Vert _{L^2}\)-norm and in special ∥⋅∥DG-norm.



This research was supported by the project GA UK No. 127615 of the Charles University (M. Balázsová) and by the grant 17-01747S of the Czech Science Foundation (M. Vlasák, who is a junior member of the University Centre for Mathematical Modeling, Applied Analysis and Computational Mathematics - MathMAC).


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic

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