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Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media
Buyang Li, Weiwei Sun
TL;DR
The paper studies whether optimal error estimates for a Galerkin-mixed FEM with a linearized semi-implicit Euler scheme require time-step restrictions. It introduces an error splitting through a time-discrete system and proves optimal L2 estimates almost without such restrictions, under stated solution assumptions, with implications for broader nonlinear parabolic systems and time discretizations.
Problem
Previous analyses imposed time-step restrictions for optimal error estimates of linearized schemes for incompressible miscible flow, motivating an analysis without such restrictions.
Method
The analysis introduces a corresponding time-discrete system and splits numerical error into temporal and spatial parts, using regularity of the time-discrete PDE solution.
Results
Optimal L2 error estimates are obtained almost without any time-step restriction for the Galerkin-mixed FEM and linearized semi-implicit Euler scheme.
Takeaways & Limitations
The error-splitting and time-discrete regularity approach is stated to extend to other nonlinear parabolic systems, higher-order methods, and other time discretizations.
Takeaways & Limitations
The analysis focuses on homogeneous boundary conditions and the lowest-order Galerkin-mixed FEM, although extensions are discussed.
Abstract
from arXiv · showhide
In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal $L^2$ error estimates hold without any time-step (convergence) condition, while all previous works require certain time-step condition. Our theoretical results provide a new understanding on commonly-used linearized schemes for nonlinear parabolic equations. The proof is based on a splitting of the error function into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.
1 Introduction
Incompressible miscible flow in porous media models coupled pressure, velocity, concentration, and diffusion processes with applications including reservoir simulation and groundwater, oil, and gas exploration. The paper addresses time-step restrictions in Galerkin-based numerical approximations by analyzing a linearized semi-implicit Euler scheme with Galerkin-mixed finite elements and splitting temporal and spatial errors.
- Problem setting: The model describes incompressible miscible flow in porous media through coupled pressure, velocity, concentration, viscosity, permeability, porosity, sources, and diffusion-dispersion terms.The domain is a bounded smooth subset of R^d with d = 2, 3, together with initial and boundary conditions.
- Motivation: The system has applications in reservoir simulations and exploration of underground water, oil, and gas.
- Related work: Earlier Galerkin-Galerkin analyses using linearized semi-implicit Euler schemes required time-step conditions such as τ = o(h), while Galerkin-mixed methods were later introduced for the system.
- Contribution: The paper analyzes a linearized semi-implicit Euler scheme with Galerkin-mixed finite elements and establishes optimal L2 error estimates almost without time-step restrictions.
- Analysis approach: The proof introduces a time-discrete system and splits numerical error into temporal-discretization and spatial-discretization components.The spatial component concerns finite element discretization of the corresponding time-discrete PDEs.
2 The Galerkin-mixed FEM and the main results
The paper formulates a linearized semi-implicit Euler scheme with Galerkin-mixed finite elements for incompressible miscible flow and states optimal L2 error results under small mesh and time-step bounds.
- Finite element spaces: The spatial discretization uses a regular triangular or tetrahedral mesh with mesh size h and associated finite element spaces.
- Assumptions: The formulation assumes regularity and structural conditions on the exact solution, viscosity, diffusion-dispersion tensor, and initial-boundary value problem.
- Time discretization: The method uses a uniform time partition with step size τ = T/N and denotes solution values at t_n by p^n, u^n, and c^n.
- Fully discrete scheme: The fully discrete mixed finite element scheme seeks discrete variables P_h^n, U_h^n, and C_h^n in the specified finite element spaces.
- Main result: Theorem 2.1 states that, for sufficiently small h and τ, the finite element system admits a unique solution satisfying the stated error estimate.
- Error analysis: The proof splits the numerical error into time-discretization error and finite element error for the corresponding time-discrete system.
3 Error estimates for time-discrete system
The time-discrete analysis defines an elliptic system, establishes existence, uniqueness, and regularity under a time-step bound, and derives estimates used in the fully discrete error analysis.
- Time-discrete system: The time-discrete solution (P^n, U^n, C^n) is defined through an elliptic system, including the concentration equation with diffusion, advection, and source terms.
- Time-discrete system: The system enforces no-flow boundary conditions for velocity and diffusive concentration flux.
- Well-posedness and regularity: The analysis proves existence, uniqueness, and regularity of the time-discrete solution when τ is smaller than a positive constant τ0.
- Well-posedness and regularity: The established regularity includes H2 bounds for pressure and velocity, W2,s bounds for concentration, time-difference bounds, and an L∞ gradient bound.
- Error estimates: The error analysis identifies the time discretization truncation error and derives estimates for the resulting error equations.
- Proof strategy: The proof combines energy estimates, elliptic regularity, Sobolev inequalities, induction, and Gronwall’s inequality.
4 Error estimates of the fully-discrete system
The fully discrete analysis uses projection-based error splitting and regularity estimates for the time-discrete PDEs to establish optimal L2 error bounds under small h and τ assumptions.
- Error-splitting framework: The analysis defines three projections to separate finite element discretization errors from errors in the corresponding time-discrete system.This projection framework underlies the fully discrete error estimates.
- Motivation: Previous Galerkin and mixed FEM analyses required an estimate that introduced a time-step restriction.The restriction could force very small time steps and increase computational cost, especially on non-uniform meshes.
- Regularity estimates: The proof establishes the auxiliary estimate (4.6) using weaker regularity of the time-discrete velocity, which is necessary for optimal L2 error estimates.The estimate follows from the regularity of (C^{n+1}, U^{n+1}) and prior error bounds.
- Conclusion of analysis: Combining the time-discrete and fully discrete estimates completes the proof of the main optimal error theorem.The analysis also estimates concentration, velocity, and pressure errors through the coupled error equations.
5 Numerical examples
Two numerical examples test the Galerkin-mixed FEM on uniform and curved-domain meshes. The reported errors exhibit the predicted spatial and temporal behavior without requiring a time-step condition.
- Experimental setup: The experiments use FreeFEM++ to solve two examples and compare numerical errors for different mesh sizes and time steps.The first example uses a unit-square domain with concentration-dependent viscosity and velocity-dependent diffusion.
- Example 5.1: With τ = 8h2, the first example reports L2 errors proportional to O(h2).The mesh is a uniform triangular partition with h = 1/M.
- Example 5.1: With fixed τ and refined h, the first example reports errors behaving like O(τ) as h/τ → 0.The authors interpret this behavior as evidence that time-step conditions are unnecessary.
- Example 5.2: The second example uses a circular domain with inhomogeneous Neumann boundary conditions and meshes with M = 32, 64, and 128 boundary points.The corresponding meshes are shown in Figure 1.
- Example 5.2: Fixed-τ computations in the second example also show that no time-step condition is needed.The errors are reported in Table 3 for several spatial mesh sizes.
6 Conclusions
The paper obtains optimal L2 error estimates for a linearized semi-implicit Euler Galerkin-mixed FEM applied to a nonlinear, strongly coupled porous-media flow system, largely without time-step restrictions.
- Conclusions: The study analyzes a nonlinear, strongly coupled parabolic system using a Galerkin-mixed FEM and a linearized semi-implicit Euler scheme.The setting is incompressible miscible flow in porous media.
- Conclusions: Optimal L2 error estimates are obtained almost without any time-step condition, unlike previous analyses that imposed restrictions on the time-step size.The proof combines error splitting with regularity analysis of the time-discrete PDEs.
- Broader scope: The error-splitting approach is presented as applicable to other nonlinear parabolic PDEs and alternative linearized or semi-implicit time discretizations.The conclusion frames this as a broader methodological possibility rather than a result proved for every such model.