Three multi-level mixed finite element methods for the steady Boussinesq equations are analyzed and discussed in this paper. The nonlinear and multi-variables coupled problem on a coarse mesh with the mesh size $h_0$ is solved firstly, and then, a series of decoupled and linear subproblems with the Stokes, Oseen and Newton iterations are solved on the successive and refined grids with the mesh sizes $h_j$, $j$ = 1, 2, . . . , $J$. The computational scales are reduced and the computational costs are saved. Furthermore, the uniform stability and convergence results in both $L^2$ - and $H^1$ -norms of are derived under some uniqueness conditions by using the mathematical induction and constructing the dual problems. Theoretical results show that the multi-level methods have the same order of numerical solutions in the $H^1$ -norm as the one level method with the mesh sizes $h_j$ = $h^2_j$−1, $j$ = 1, 2, . . . , $J$. Finally, some numerical results are provided to investigate and compare the effectiveness of the multi-level mixed finite element methods.

A local parallel fully mixed finite element method for superposed fluid and porous layers

Hybridizable H ( d i v ) H(div) -conforming finite elements for symmetric tensors on simplices with barycentric refinement are developed in this work for arbitrary dimensions and any polynomial order. By employing barycentric refinement and an intrinsic tangential-normal decomposition, novel basis functions are constructed to redistribute degrees of freedom while preserving H ( d i v ) H(div) -conformity and symmetry, and ensuring inf-sup stability. These hybridizable elements enhance computational flexibility and efficiency, with applications to mixed finite element methods for linear elasticity.

Postprocessing mixed finite element methods for the Cahn–Hilliard equation: the fully discrete case

Superconvergence analysis of low-order nonconforming mixed finite element method for the time-dependent incompressible MHD equations

A nonstationary mixed hemivariational inequality is studied for an incompressible fluid flow described by the Stokes equations subject to a nonsmooth boundary condition of friction type described by the Clarke subdifferential. The solution existence is shown through a limiting procedure based on temporally semi-discrete approximations. Uniqueness of the solution and its continuous dependence on data are also established. Fully discrete numerical methods are introduced to solve the nonstationary mixed hemivariational inequality. The backward Euler scheme is applied to discretize the time derivative, and mixed finite element methods are used for the spatial discretization. An error bound is derived for the numerical solution of the unknown velocity. Numerical results are reported on computer simulations of some examples.

Abstract The optimal $L^{2}$-error estimates of a locking-free numerical method are established for a quasi-static linear poroelasticity. Existing research shows that the original model can be reformulated as a generalized Stokes equation coupled with a diffusion problem, which inherently mitigates two locking phenomena associated with the continuous Galerkin method. However, previous studies only achieved optimal error estimates in the $H^{1}$-norm, lacking results for displacement in the $L^{2}$-norm. We find that the main barrier to achieve these estimates is arising from the influence of Lamé constant $\lambda $ on numerical schemes. By taking the value of $\lambda $ into account, we design a fully discrete mixed finite element method to solve the reformulated problem. We prove that the discrete formulation satisfies the inf-sup stability condition across various finite element pairs. Our results show that when $\lambda <\infty,$ the generalized Stokes problem is stable with equal-order Lagrange element pairs, making the Taylor–Hood pairs unnecessary and enabling optimal $L^{2}$-error estimates for displacement. For $\lambda \rightarrow \infty,$ the problem reduces to a standard Stokes problem. Using inf-sup bounds and the Aubin–Nitsche duality method, we provide optimal $L^{2}$-error analysis. Numerical examples are included to confirm our theoretical findings.

Mixed finite element method with Bernstein elements for the 2D steady Navier-Stokes equations: Stability and implementation

A new stabilizer-free weak Galerkin mixed finite element method for the biharmonic equation on polygonal meshes

Summary Hydraulic fracturing remains a dynamic and evolving research field, with laboratory investigations continually revealing phenomena that challenge conventional modeling tools. Despite advances in simulation techniques, key limitations persist—particularly in representing fracture branching and the influence of distinct fracturing fluids. These complexities are well-suited to phase-field approaches, which replace sharp crack discontinuities with diffusive damage zones, enabling the modeling of intricate crack geometries. By incorporating surface energy—known to vary in the presence of different fluids—phase-field models also capture fluid-specific effects. Although the method introduces additional mathematical constructs, its foundations remain deeply rooted in Griffith’s classical energetic framework for crack propagation. Resolving narrow diffusive zones demands fine mesh resolution, resulting in significant computational overhead. The present work introduces a computational framework that accelerates phase-field simulations via dynamic adaptive gridding (DAG), concentrating refinement in regions of active fracture propagation. A low-pressure condition is imposed on newly formed cracks to reproduce pressure responses consistent with experimental studies. DAG is implemented on fully unstructured triangular grids, recursively refining elements while preserving child-triangle geometry. Hanging nodes generated during refinement are systematically removed to ensure compatibility with the numerical method. Newly formed cracks are identified via the phase-field variable, and affected cells are assigned low-pressure values in accordance with experimental observations. The mixed hybrid finite element method (MHFEM) is used to solve for pressure, guaranteeing local mass conservation. Phase-field evolution is also addressed with MHFEM, while mechanical deformation is solved using conventional finite element method (FEM). Fluid properties are characterized using equations sourced from National Institute of Standards and Technology databases, and the coupled system of three equations is solved using a sequentially iterative scheme. Speedups are computed across two laboratory-scale domains and one intermediate-scale example, providing insight into future field-scale developments. The DAG approach shows pronounced benefits at larger scales, yielding orders-of-magnitude improvements in computational efficiency. The characteristic pressure history—buildup, rapid decay, and slow propagation—is recovered with the imposed low-pressure condition. Few implementations of DAG exist for phase-field hydraulic fracturing models, and prior efforts are generally restricted to structured meshes and open-source libraries. This work also avoids the negative pressures at the fracture tip reported in other approaches.

Numerical Analysis of a Second Order Crank-Nicolson Mixed Finite Element Method for the Swift-Hohenberg Equation

A mixed finite element method for pricing American options and Greeks in the Heston model

ABSTRACT In this paper, the ‐Galerkin mixed finite element method (MFEM) is used to solve the time‐fraction‐order damped beam vibration equations with simple support at both ends. Compared with the standard finite element method (FEM), the ‐Galerkin MFEM can calculate the deflection, bending moment and other parameters of the beam more directly, so it is more suitable for solving the beam vibration equations, which are high‐order partial differential equations. The Caputo fractional derivative is approximated using the L1 formula, and a fully discrete numerical scheme is established by linear backward Euler method. The stability of the proposed scheme, as well as the existence, uniqueness, and convergence of the numerical solution, are rigorously analyzed. Numerical examples are provided to verify the theoretical results, with real‐world beam data used to investigate how variations in the damping coefficient and the order of the fractional derivative affect the beam's vibrational behavior. Numerical simulations demonstrate that the amplitude of the beam decays more rapidly with larger damping coefficients. Moreover, as the order of the fractional derivative increases, the decay rate of the beam vibration first increases and then decreases, while the peak of the curve gradually shifts to the right.

In this paper, we consider mixed finite element semi-/full discretizations of the Rosensweig ferrofluid flow model. We first establish some regularity results for the model under several basic assumptions. Then we show that the energy stability of the weak solutions is preserved exactly for both the semi- discrete and fully discrete finite element solutions. Moreover, we give existence and uniqueness results and derive optimal error estimates for the discrete schemes. Finally, we provide numerical experiments to verify the theoretical results.

Observability Inequalities of a Semi-discrete Schrödinger Integro-Differential Equation Derived from a Mixed Finite Element Method

Conservative mixed Discontinuous Galerkin finite element method with full decoupling strategy for incompressible MHD problems

The Kirsch problem, namely the problem of an infinite plane with a circular hole under uniaxial tension, is one of the cornerstone problems in elasticity. The Kirsch problem is rooted in classical elasticity theory, which cannot explain the size effects. We investigate the Kirsch problem for the first time in the framework of Mindlin’s second strain gradient elasticity theory. By presenting the fundamental equations in polar coordinates, we use the stress function to derive the closed-form solution of the Kirsch problem. Then, we compare the solution to its counterparts associated with first strain gradient elasticity and classical elasticity. Our results indicate that the strain gradient effects can increase stiffness and reduce the stress concentration around the hole. Under the second strain gradient elasticity theory, a larger circular hole radius results in higher stress concentration, exhibiting a significant size effect. Furthermore, we establish a mixed finite-element method for second strain gradient elasticity. The results of computational simulations are in close agreement with the solution proposed in this contribution. This work extends the Kirsch problem to second strain gradient elasticity providing a benchmark solution highlighting the importance of higher gradient effects in predicting the elastic behaviour of materials at smaller scales.

A conforming and mass conservative pseudostress-based mixed finite element method for the stationary Stokes problem

We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (0<α<1) with mixed finite element methods (P1b–P1 and P2–P1) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term γ|u|r−2u, for r≥2. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models.

New Fully Mixed Finite Element Methods for the Coupled Convective Brinkman-Forchheimer and Nonlinear Transport Equations