scholarly journals Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

2018 ◽  
Vol 19 (2) ◽  
pp. 333-374 ◽  
Author(s):  
Jérôme Droniou ◽  
Robert Eymard ◽  
Alain Prignet ◽  
Kyle S. Talbot
Keyword(s):  
Convergence Analysis ◽  
Miscible Displacement ◽  
Numerical Schemes ◽  
Displacement Problem

scholarly journals Convergence analysis of time-discretisation schemes for rate-independent systems

2019 ◽  
Vol 25 ◽  
pp. 65 ◽  
Author(s):  
Dorothee Knees
Keyword(s):  
Convergence Analysis ◽  
Numerical Schemes ◽  
Solution Concepts ◽  
Continuous Solutions ◽  
Energy Functionals ◽  
Convergence Of Solutions ◽  
Time Discretisation ◽  
Energetic Solutions ◽  
Nonconvex Energy ◽  
The Given

It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example.

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