Estimates of Discrete Time Derivatives for the Parabolic-Parabolic Robin-Robin Coupling Method

Core Concepts

The authors prove second-order convergence estimates for the discrete time derivatives of the scalar fields in the Robin-Robin coupling method applied to a parabolic-parabolic interface problem. They also establish H2 error estimates in a special case where the interface is flat and perpendicular to two sides of the domain.

Abstract

The content discusses a Robin-Robin coupling method for solving a parabolic-parabolic interface problem. The key points are: The authors consider a parabolic-parabolic interface problem with continuity of both the primal variables and the fluxes across the interface. They analyze a loosely coupled, non-iterative Robin-Robin coupling method for this problem and prove estimates for the discrete time derivatives of the scalar fields in different norms. When the interface is flat and perpendicular to two of the edges of the domain, they prove error estimates in the H2-norm, which are important for analyzing a defect correction method. Numerical results are provided to support the theoretical findings. The main motivation for this work is to establish the necessary estimates for analyzing a prediction-correction method that aims to improve the first-order convergence of the splitting method to second-order convergence in time.

Stats

∆t N−1 X n=0 νf∥D2(U n+1)∥2 L2(Ωf) ≤CΞN(∂∆th1, ∂∆th2, ∂∆t(αg1 −g2), ∂∆t˜ g2) CΞN(∂xh1, ∂xh2, ∂x(αg1 −g2), ∂∆t∂x˜ g2) C∆t N−1 X n=0 νf∥hn+1 1 ∥2 L2(Ωf) ∆t N−1 X n=0 νf∥D2(∂∆tU n+1)∥2 L2(Ωf ) ≤CΞN(∂2 ∆th1, ∂2 ∆th2, ∂2 ∆t(αg1 −g2), ∂2 ∆t˜ g2) CΞN(∂∆t∂xh1, ∂∆t∂xh2, ∂∆t∂x(αg1 −g2), ∂∆t∂x˜ g2) C∆t N−1 X n=0 νf∥∂∆thn+1 1 ∥2 L2(Ωf)

Quotes

None

Key Insights Distilled From

Estimates of discrete time derivatives for the parabolic-parabolic Robin-Robin coupling method

by Erik... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01594.pdf

Deeper Inquiries

How can the analysis be extended to the fully discrete case, where both spatial and temporal discretizations are considered

To extend the analysis to the fully discrete case, we need to consider both spatial and temporal discretizations. This involves discretizing the spatial domain into elements and applying numerical methods to solve the differential equations at each time step within each element. For the spatial discretization, finite element methods or finite difference methods can be used to approximate the solution within each element. The temporal discretization involves discretizing the time domain into time steps and applying time-stepping methods such as the backward Euler method or Crank-Nicolson method to evolve the solution in time. By combining the spatial and temporal discretizations, we can analyze the error estimates and convergence rates of the fully discrete method, similar to what was done for the semi-discrete case.

What are the implications of the H2 error estimates for the design and analysis of defect correction methods for parabolic-parabolic interface problems

The H2 error estimates obtained in the analysis have significant implications for the design and analysis of defect correction methods for parabolic-parabolic interface problems. These error estimates provide insights into the convergence rates of the numerical solution and the accuracy of the method in approximating the true solution. In the context of defect correction methods, the H2 error estimates can be used to assess the effectiveness of the correction step in improving the convergence rate from first order to second order in time. By understanding the error behavior in stronger norms like H2, researchers can optimize the defect correction process to achieve higher accuracy and efficiency in solving parabolic-parabolic interface problems.

Can the techniques developed in this work be applied to other types of interface problems, such as fluid-structure interaction or wave-parabolic couplings

The techniques developed in this work can potentially be applied to other types of interface problems, such as fluid-structure interaction or wave-parabolic couplings. The analysis of discrete time derivatives and error estimates in different norms is a fundamental aspect of numerical methods for interface problems. By adapting the methods and error analysis techniques to different types of interface problems, researchers can gain insights into the convergence behavior, stability, and accuracy of numerical algorithms in various applications. The principles and methodologies established in this work can serve as a foundation for studying and solving a wide range of interface problems in computational mathematics and engineering.