Optimal and Superconvergence Error Estimates for the Div Least-Squares Finite Element Method on Elliptic Problems

Centrala begrepp

The paper presents a complete error analysis for the div least-squares finite element method on elliptic problems, improving the current state-of-the-art results. Optimal error estimates and superconvergence results are established for both the scalar and flux variables, often without requiring higher regularity assumptions.

Sammanfattning

The paper discusses the error estimations for the div least-squares finite element method (div-LSFEM) on elliptic problems. The key highlights and insights are:

1. The authors present a complete error analysis for div-LSFEM, which improves the current state-of-the-art results.

2. The error estimations for both the scalar and flux variables are established using dual arguments. In most cases, only an H^(1+ε) regularity is required.

3. Optimal error estimates are obtained without any higher regularity assumption in many cases, including the common choices of k=m and k+1=m, where k and m are the degrees of the flux and scalar finite element spaces, respectively.

4. Superconvergence results are derived, showing improved convergence rates for certain projections of the solution.

5. Numerical experiments are provided to confirm the theoretical analysis.

The authors consider the model problem (1.1) and introduce the first-order system (1.3) to apply the div-LSFEM. The primary error estimates are obtained in Theorem 3.9 using projections. Further superconvergence results are derived in Section 4 by dual arguments.

Statistik

∥u - u_h∥_0 + ∥q - q_h∥_0 ≤ Ch^(k+1)(∥u∥_(k+1) + ∥q∥_(k+1)), for convex Ω and k=m
∥∇(Π_V^h u - u_h)∥_0 + ∥∇·(Π_P^h q - q_h)∥_0 ≤ Ch^(k+1)(∥u∥_(k+1) + ∥q∥_(k+1)), for convex Ω and k=m
∥u - u_h∥_0 ≤ Ch^(m+1)(∥u∥_(m+1) + ∥q∥_m), for convex Ω and m=k+1
∥q - q_h∥_0 ≤ Ch^(k+1)(∥u∥_(k+2) + ∥q∥_(k+1)), for convex Ω and m=k+1
∥∇·(Π_P^h q - q_h)∥_0 ≤ Ch^(k+1)(∥u∥_(k+2) + ∥q∥_(k+1)), for convex Ω, ω=0, and Ph=BDM_k (k≥2) or Ph=RT_k (k≥1)
∥∇·(Π_P^h q - q_h)∥_0 ≤ Ch^2(∥u∥_2 + ∥∇·q∥_1 + ∥q∥_1), for Ph=RT_0 and Vh=P_m

Citat

"The error estimations for both the scalar and the flux variables are established by dual arguments, and in most cases, only an H^(1+ε) regularity is used."
"Numerical experiments strongly confirm our analysis."

Viktiga insikter från

Superconvergence error estimates for the div least-squares finite element method on elliptic problems

by Gang Chen,Fa... på arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04918.pdf

Djupare frågor

How can the div-LSFEM be extended to more general partial differential equations beyond the elliptic problem considered in this work

The div-LSFEM can be extended to more general partial differential equations by considering different types of problems and adjusting the formulation accordingly. For instance, the methodology can be applied to problems involving different boundary conditions, non-linear terms, or variable coefficients in the differential equations. By modifying the bilinear forms and introducing appropriate projections, the div-LSFEM can be adapted to handle a wider range of PDEs, such as parabolic or hyperbolic equations. Additionally, the technique can be extended to three-dimensional domains and more complex geometries by incorporating suitable discretization methods and basis functions.

What are the potential limitations or challenges in applying the superconvergence results derived in this paper to practical engineering problems

While the superconvergence results derived in the paper provide valuable insights into the accuracy and efficiency of the div-LSFEM, there are potential limitations and challenges in applying these results to practical engineering problems. One limitation could be the complexity of real-world problems compared to the simplified model considered in the analysis. Practical problems often involve uncertainties, non-linearity, and heterogeneous materials, which may affect the applicability of the superconvergence results. Additionally, the computational cost and implementation challenges of achieving superconvergence in realistic simulations could pose practical limitations. It is essential to validate the superconvergence results in practical engineering scenarios to ensure their effectiveness and reliability.

Can the techniques developed in this paper be adapted to analyze other types of mixed finite element methods for elliptic problems

The techniques developed in this paper can be adapted to analyze other types of mixed finite element methods for elliptic problems by incorporating similar error estimation strategies and dual arguments. By applying the principles of least-squares finite element methods and utilizing projections to approximate the solutions, the analysis can be extended to different mixed formulations, such as the Raviart-Thomas or Brezzi-Douglas-Marini elements. The key lies in understanding the specific properties and requirements of the mixed finite element method under consideration and adapting the error analysis framework accordingly. By leveraging the insights and methodologies presented in this paper, researchers can explore the application of superconvergence techniques to a broader class of mixed finite element methods for elliptic problems.