arXiv subjects

Yanren Hou

Publications and source records attributed to Yanren Hou.

7 recordsLinked to original sources

arXiv2025

A priori error estimates for stable generalized finite element discretization of parabolic interface optimal control problems

In this paper, we investigate optimal control problems governed by the parabolic interface equation, in which the control acts on the interface. The solution to this problem exhibits low global regularity due to the jump of the coefficient across the interface and the control acting on the interface. Consequently, the traditional finite element method fails to achieve optimal convergence rates when using a uniform mesh. To discretize the problem, we use fully discrete approximations based on the stable generalized finite element method for spatial discretization and the backward Euler scheme for temporal discretization, as well as variational discretization for the control variable. We prove a priori error estimates for the control, state, and adjoint state. Numerical examples are provided to support the theoretical findings.

math.NA↗

arXiv2025

Numerical analysis of an H(div)-conforming divergence-free DG method with a second-order explicit Runge-Kutta scheme for incompressible flows

Recently, H(div)-conforming DG type methods coupled with Runge-Kutta (RK) time stepping have been widely employed for simulating high Reynolds number flows, with the convective terms treated explicitly. Although the analysis techniques of RKDG methods were well developed, the extension to incompressible flows is highly nontrivial due to the exactly divergence-free constraint, where the key lies in analyzing the convective terms. We neglect viscosity effects, and conduct an error analysis for an H(div)-conforming divergence-free DG method combined with a second-order explicit RK scheme, for the incompressible Euler equations. We derive an a priori error estimate of $O(h^{k+1 / 2}+\tau^2)$ under a restrictive CFL condition $\tau \lesssim h^{4 / 3}$ for polynomials of degree $k \geq 1$, where $h$ and $\tau$ are the mesh size and time step size, respectively, assuming that the exact solution is smooth. For the case of linear polynomials, we investigate whether existing analytical techniques can relax the restrictive CFL condition to a standard CFL condition $\tau \lesssim h$. It is demonstrated that the exactly divergence-free constraint prevents the application of these techniques. We conjecture that the error estimates for linear polynomials cannot be derived under a standard CFL condition. Finally, we mention that based on our analytical framework, our analytical results will be readily extended to the Navier-Stokes equations at high mesh Reynolds number, with the viscous and convective terms treated explicitly. Numerical experiments are conducted, supporting our analytical results and the conjecture for linear polynomials.

math.NA↗

arXiv2020

Expandable Local and Parallel Two-Grid Finite Element Scheme for the Stokes Equations

In this paper, we present a novel local and parallel two-grid finite element scheme for solving the Stokes equations, and rigorously establish its a priori error estimates. The scheme admits simultaneously small scales of subproblems and distances between subdomains and its expansions, and hence can be expandable. Based on the a priori error estimates, we provide a corresponding iterative scheme with suitable iteration number. The resulting iterative scheme can reach the optimal convergence orders within specific two-grid iterations ($O(|\ln H|^2)$ in 2-D and $O(|\ln H|)$ in 3-D) if the coarse mesh size $H$ and the fine mesh size $h$ are properly chosen. Finally, some numerical tests including 2-D and 3-D cases are carried out to verify our theoretical results.

math.NA↗

arXiv2020

A variable timestepping algorithm for the unsteady Stokes/Darcy model

This report considers a variable step time discretization algorithm proposed by Dahlquist, Liniger and Nevanlinna and applies the algorithm to the unsteady Stokes/Darcy model. Although long-time forgotten and little explored, the algorithm performs advantages in variable timestep analysis of various fluid flow systems, including the coupled Stokes/Darcy model. The paper proves that the approximate solutions to the unsteady Stokes/Darcy model are unconditionally stable due to the G-stability of the algorithm. Also variable time stepping error analysis follows from the combination of G-stability and consistency of the algorithm. Numerical experiments further verify the theoretical results, demonstrating the accuracy and stability of the algorithm for time-dependent Stokes/Darcy model.

math.NA↗

arXiv2016

On the Weak Solutions to Mixed Navier-Stokes-Darcy Model

In this paper, an a priori estimate of weak solutions to the mixed Navier-Stokes/Darcy model with Beavers-Joseph-Saffman's interface condition and the existence of a weak solution are established without the small data and/or the large viscosity restriction for the first time. Based on these results, the global uniqueness of the weak solution is obtained.

math.AP↗

arXiv2015

An Expandable Local and Parallel Two-Grid Finite Element Scheme

An expandable local and parallel two-grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of Poisson equation. Compared with the usual local and parallel finite element schemes, the scheme proposed in this paper can be easily implemented in a large parallel computer system that has a lot of CPUs. Convergence results base on $H^1$ and $L^2$ a priori error estimation of the scheme are obtained, which show that the scheme can reach the optimal convergence orders within $|\ln H|^2$ or $|\ln H|$ two-grid iterations if the coarse mesh size $H$ and the fine mesh size $h$ are properly configured in 2-D or 3-D case, respectively. Some numerical results are presented at the end of the paper to support our analysis.

math.NA↗

arXiv2015

Optimal Error Estimates of A Decoupled Scheme Based on Two-Grid Finite Element for Mixed Stokes-Darcy Model

Although the numerical results suggest the optimal convergence order of the two-grid finite element decoupled scheme for mixed Stokes-Darcy model with Beaver-Joseph-Saffman interface condition in literatures, the numerical analysis only get the optimal error order for porous media flow and a non-optimal error order that is half order lower than the optimal one in fluid flow. The purpose of this paper is to fill in the gap between the numerical results and the theoretical analysis. By introducing an $H^1-$ orthogonal decomposition of a specific vector valued space, we obtain the optimal error estimates of the velocity and pressure in fluid flow region.

math.NA↗