Comparison of Finite Element and Finite Difference Methods for Elliptic Cross-Interface Problems: Highlighting Discrepancies in Solutions for High-Contrast, High-Frequency Coefficients

Core Concepts

While finite element and finite difference methods produce similar solutions for elliptic cross-interface problems with low-frequency coefficient oscillations, significant discrepancies arise in scenarios involving high-contrast and high-frequency coefficient functions, a phenomenon exemplified by the SPE10 benchmark problem.

Abstract

• Bibliographic Information: Feng, Q. (2024). Distinct Numerical Solutions for Elliptic Cross-Interface Problems Using Finite Element and Finite Difference Methods. arXiv preprint arXiv:2408.10459v3.
• Research Objective: This paper investigates the performance of second-order finite element (FEM) and finite difference (FDM) methods in solving elliptic cross-interface problems characterized by piecewise constant coefficients with high contrast and high-frequency oscillations.
• Methodology: The study employs a simplified 2D elliptic cross-interface problem with interfaces intersecting along vertical and horizontal straight lines. Uniform Cartesian grids are used for both FEM and FDM implementations. Numerical experiments are conducted with varying coefficient functions, including cases with high contrast and low-frequency oscillations, low contrast and high-frequency oscillations, and high contrast and high-frequency oscillations.
• Key Findings: The research reveals that FEM and FDM yield similar numerical solutions when the coefficient functions exhibit either high contrast with low-frequency oscillations or low contrast with high-frequency oscillations. However, significant differences in numerical solutions are observed when high-contrast and high-frequency coefficient functions are employed. This discrepancy is particularly noteworthy in problems like the SPE10 benchmark, which involves high-contrast and high-frequency permeability due to varying geological layers.
• Main Conclusions: The study highlights the potential for significant discrepancies between FEM and FDM solutions in elliptic cross-interface problems with high-contrast and high-frequency coefficients. This finding has important implications for developing multiscale methods, where reference solutions are typically obtained using FEM with fine meshes.
• Significance: This research underscores the need for careful consideration when choosing numerical methods for elliptic cross-interface problems, particularly in applications involving high-contrast and high-frequency coefficients. The findings have implications for fields like computational fluid dynamics, porous media flow simulation, and multiscale modeling.
• Limitations and Future Research: The study focuses on a simplified 2D problem. Further research is needed to investigate the observed discrepancies in more complex scenarios, including 3D problems and those with more general interface geometries. Exploring alternative numerical methods specifically designed for high-contrast, high-frequency interface problems is also crucial.

Stats

The SPE10 benchmark problem typically involves high-contrast and high-frequency permeability.
When the coefficient functions exhibit either high jumps with low-frequency oscillations or low jumps with high-frequency oscillations, the finite element method and finite difference method yield similar numerical solutions.
When the interface problems involve high-contrast and high-frequency coefficient functions, the numerical solutions obtained from the finite element and finite difference methods differ significantly.

Quotes

"However, our numerical results show that FEM and FDM produce markedly different solutions for elliptic intersecting interface problems with high-contrast and highly oscillatory coefficient functions (see Figs. 9 to 12 in Examples 4 to 7)."
"Given that the widely studied SPE10 benchmark problem (see https://www.spe.org/web/csp/datasets/set02.htm) typically involves high-contrast and high-frequency permeability due to varying geological layers in porous media, this phenomenon warrants attention."
"To our best knowledge, so far there is no available literature that has clearly observed such significant differences in the numerical solutions produced by finite element and finite difference methods."

Key Insights Distilled From

Distinct Numerical Solutions for Elliptic Cross-Interface Problems Using Finite Element and Finite Difference Methods

by Qiwei Feng at arxiv.org 11-04-2024

https://arxiv.org/pdf/2408.10459.pdf

Deeper Inquiries

How can the observed discrepancies between FEM and FDM solutions for high-contrast, high-frequency elliptic cross-interface problems be mitigated or accounted for in practical applications?

Several strategies can be employed to mitigate or account for the discrepancies between FEM and FDM solutions in high-contrast, high-frequency elliptic cross-interface problems:
1. Mesh Refinement and Adaptive Meshing:

For FDM: Refining the mesh size, particularly near the interfaces where the coefficient function exhibits high-frequency oscillations or discontinuities, can improve the accuracy of FDM solutions. However, this approach can significantly increase computational cost.
For FEM: Adaptive mesh refinement strategies, where the mesh is selectively refined in regions of high solution gradients or coefficient variation, can be more computationally efficient while improving accuracy.
2. Higher-Order Schemes:

For FDM: Employing higher-order finite difference stencils, such as fourth-order or sixth-order schemes, can better capture the solution behavior in regions of high variation. However, these schemes often require special treatment near boundaries and interfaces, adding complexity to the implementation.
For FEM: Utilizing higher-order basis functions within the FEM framework can lead to improved accuracy with coarser meshes.
3. Specialized Methods for Interface Problems:

Immersed Interface Methods (IIM): IIM embed the interface onto a fixed Cartesian grid and modify the finite difference stencils near the interface to account for the jump conditions. This approach can achieve high-order accuracy even with complex interfaces.
Matched Interface and Boundary (MIB) Methods: MIB methods use fictitious grid points and high-order interpolation to enforce the jump conditions at the interface. These methods can also handle complex geometries and achieve high-order accuracy.
Discontinuous Galerkin (DG) Methods: DG methods allow for discontinuities in the solution across element boundaries, making them well-suited for interface problems with discontinuous coefficients. They offer flexibility in handling complex geometries and can achieve high-order accuracy.
4. Solution Correction Techniques:

Post-processing techniques: These methods aim to improve the accuracy of an existing solution (obtained from either FEM or FDM) by applying local corrections near the interface.
5. Benchmarking and Validation:

It is crucial to validate numerical solutions against analytical solutions whenever possible. In cases where analytical solutions are unavailable, benchmark problems with known solutions can be used to assess the accuracy of different numerical methods.
Practical Considerations:
The choice of the most appropriate strategy depends on factors such as:

Accuracy requirements: Higher accuracy demands may necessitate more sophisticated methods or finer meshes.
Computational cost: Mesh refinement and higher-order schemes increase computational cost, while specialized methods may require more complex implementations.
Complexity of the interface: Complex interfaces may favor methods specifically designed for such geometries.

Could the differences in numerical solutions stem from the inherent properties of FEM and FDM, such as their treatment of boundary conditions or their approximation of derivatives, rather than solely from the high-contrast and high-frequency nature of the coefficient function?

Yes, the observed discrepancies can also arise from the inherent differences between FEM and FDM, particularly in how they handle boundary conditions and approximate derivatives:
1. Treatment of Boundary Conditions:

FDM: Finite difference methods typically require boundary conditions to be explicitly discretized at grid points. This can lead to accuracy issues, especially for complex boundary conditions or curved boundaries.
FEM: Finite element methods incorporate boundary conditions weakly through the variational formulation. This often results in a more accurate representation of boundary conditions, even for complex geometries.
2. Approximation of Derivatives:

FDM: Finite difference methods approximate derivatives using finite difference stencils, which can be less accurate in regions of high solution gradients or coefficient variation.
FEM: Finite element methods represent the solution as a linear combination of basis functions, and derivatives are computed based on these basis functions. This approach can provide a more accurate approximation of derivatives, particularly when using higher-order basis functions.
3. Handling High-Contrast Coefficients:

FDM: Standard finite difference stencils can lead to significant errors near interfaces with high-contrast coefficients. This is because the stencils assume a smooth solution, which is not the case across such interfaces.
FEM: While FEM can handle discontinuities in the coefficient function better than FDM, high-contrast coefficients can still lead to ill-conditioned matrices, affecting the accuracy and stability of the solution.
In summary: While high-contrast and high-frequency coefficient functions exacerbate the discrepancies, the inherent differences in how FEM and FDM treat boundary conditions and approximate derivatives contribute to the observed differences in numerical solutions.

If the accuracy of both FEM and FDM is compromised in these specific scenarios, what alternative numerical approaches could be explored to provide more reliable solutions for elliptic cross-interface problems with high-contrast and high-frequency coefficients, especially in fields like computational fluid dynamics and material science where such problems frequently arise?

Given the limitations of standard FEM and FDM in scenarios involving high-contrast and high-frequency coefficients, several alternative numerical approaches can be considered for more reliable solutions to elliptic cross-interface problems:
1. Extended Finite Element Method (XFEM):

XFEM enhances the standard FEM by incorporating additional basis functions that capture the discontinuities and singularities associated with interfaces. This enables accurate representation of the solution even with high-contrast coefficients across interfaces.
2. Generalized Finite Element Method (GFEM):

GFEM enriches the FEM approximation space with special functions that capture the local behavior of the solution. This method is particularly effective for problems with singularities or high gradients near interfaces.
3. hp-Finite Element Method:

The hp-FEM combines mesh refinement (h-refinement) with the use of higher-order polynomials (p-refinement) within each element. This adaptive approach allows for efficient resolution of both smooth and non-smooth solution features, making it suitable for high-contrast and high-frequency problems.
4. Finite Volume Method (FVM):

FVM is a conservative method that discretizes the integral form of the governing equations over control volumes. It handles discontinuities in the coefficient function naturally and is well-suited for conservation laws often encountered in computational fluid dynamics.
5. Boundary Element Method (BEM):

BEM reformulates the problem as an integral equation over the boundary of the domain. This method is particularly advantageous for problems where the solution is primarily of interest on the boundary or in the presence of infinite domains.
6. Meshless Methods:

Meshless methods, such as the Element-Free Galerkin (EFG) method and the Reproducing Kernel Particle Method (RKPM), avoid the need for a mesh, offering flexibility in handling complex geometries and discontinuities.
7. Multiscale Methods:

Multiscale methods, like the Heterogeneous Multiscale Method (HMM) and the Multiscale Finite Element Method (MsFEM), aim to capture the effects of fine-scale variations in the coefficient function on the coarse-scale solution. These methods are computationally efficient for problems with high-frequency oscillations in the coefficients.
Field-Specific Applications:

Computational Fluid Dynamics: FVM, XFEM, and multiscale methods are commonly employed for fluid flow problems with high-contrast properties, such as flow in porous media with varying permeability.
Material Science: XFEM, GFEM, and hp-FEM are well-suited for problems involving material interfaces, cracks, and dislocations, where high stress concentrations and discontinuities occur.
The selection of the most appropriate method depends on the specific problem characteristics, accuracy requirements, and computational resources available.