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Full Discretisations for Nonlinear Evolutionary Inequalities Based on Stiffly Accurate Runge-Kutta and hp-Finite
1Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany.
Foundations of Computational Mathematics (New York, N.Y.)
|June 2, 2015
Summary
This study analyzes the convergence of numerical methods for nonlinear evolutionary inequalities. It establishes a convergence result for time-discretized approximations, crucial for solving complex differential inclusions.
Area of Science:
- Numerical Analysis
- Partial Differential Equations
- Computational Mathematics
Background:
- Nonlinear evolutionary inequalities are central to modeling complex phenomena.
- Implicit Runge-Kutta and nonconforming Galerkin methods are powerful numerical tools.
- Understanding their convergence is vital for accurate simulations.
Purpose of the Study:
- To investigate the convergence of fully discretized nonlinear evolutionary inequalities.
- To analyze the application of implicit Runge-Kutta and nonconforming Galerkin methods.
- To establish convergence for piecewise constant time interpolants.
Main Methods:
- Utilizing implicit Runge-Kutta (IRK) and nonconforming Galerkin methods.
- Applying set convergence (Glowinski-Mosco-Stummel) for unilateral constraints.
- Formulating fully discrete variational inequalities based on Signorini problems.
Main Results:
- Demonstrated convergence for a class of stiffly accurate IRK methods (Radau IIA, Lobatto IIIC).
- Established convergence of the piecewise constant in time interpolant.
- Provided a convergence result under hypotheses related to existence theory.
Conclusions:
- The study successfully demonstrates the convergence of combined numerical methods for nonlinear evolutionary inequalities.
- This work advances the numerical treatment of differential inclusions with monotone operators.
- The findings are applicable to problems like Signorini-type initial-boundary value problems.
Keywords:
ConvergenceMonotone operatorsNonconforming Galerkin methodsNonlinear differential inclusionsNonlinear evolutionary inequalitiesStabilityStiffly accurate Runge–Kutta methodshp-finite element approximationsMore Related Videos
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