Stability and convergence analysis of mixed finite element approximations for a Biot-Brinkman model

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

A Macro–Microscopic Coupled Mixed Finite-Element Model for One-Dimensional Nonlinear Consolidation

High-precision nonlinear comprehensive analysis of curved beams with shear effects under extreme loads via advanced mixed finite elements

Abstract. Rocks break if shear stresses exceed their strength. It is therefore important for typical geoscientific applications to take shear failure mechanism and the subsequent development of mode-II shear bands or faults into account. Many existing codes incorporate non-associated Drucker-Prager or Mohr-Coulomb plasticity models to simulate this behavior. Yet, when effective mean stress becomes extensional, for example when fluid pressure becomes large, the dominant failure mode changes to a mode-I (opening) mode, which initiates plastic volumetric deformation. It is rather difficult to represent both failure modes in numerical models in a self-consistent manner, while also accounting for the nonlinear visco-elastic host rock rheology, which varies from being nearly incompressible in the mantle to being compressible in surface-near regions. Here, we present a simple plasticity model that is designed to overcome these difficulties. We employ a combination of a linearized Drucker-Prager shear failure envelope with a circular tensile cap function in way that ensures continuity and smoothness of both yield surface and flow potential in the entire stress space. A Perzyna-type viscoplastic regularization ensures that the resulting localization zones are mesh-insensitive. To deal with the near incompressibility condition, a mixed two-field finite element formulation is employed. The local nonlinear iterations at the integration-point level are used to determine the stress increments. The global Newton-Raphson iterations are applied to solve the discretized momentum and continuity residual equations. The presented plasticity model is implemented in an open-source 2D unstructured finite element code GeoTech2D. The results of several typical test cases that range from crustal scale deformation to the propagation of fluid-induced tensile failure zones demonstrate rapid convergence. The robustness of the solution scheme is enhanced by the adaptive time stepping algorithm.

Maxillary central incisor implant restoration in relation to the adjacent natural teeth: A biomechanical analysis.

A new two-grid mixed finite element Crank-Nicolson method for the temporal fractional fourth-order sine-Gordon equation

ABSTRACTThis paper presents an efficient computational framework for solving convection‐diffusion obstacle problems, designed for convection‐dominated regimes while ensuring local and global mass conservation. The method relies on an operator‐splitting strategy that decouples the problem into convection and diffusion sub‐problems, treated, respectively, in Lagrangian and Eulerian settings. The convective transport is handled by a particle‐in‐cell method, while the diffusion, formulated as a parabolic variational inequality, is discretized using mixed finite elements. This leads to symmetric saddle‐point systems with complementarity conditions, solved efficiently via a primal‐dual active set algorithm. To ensure conservative coupling between particles and mesh, a PDE‐constrained projection is employed. The effectiveness and performance of the overall approach have been established by rigorous benchmarks with analytical solutions from the literature, covering both structured and unstructured meshes.

Аддитивные технологии открывают новые возможности для создания персонализированных имплантатов, обладающих неоднородной структурой. Численный анализ таких изделий сопровождается определенными трудностями, связанными с нерегулярностью исследуемой геометрии, локальными перепадами толщин и сложной пространственной топологией, обусловленной микроструктурными особенностями, что требует разработки специализированных подходов к дискретизации. Это является мотивацией к построению моделей из элементов различного типа, обеспечивающих баланс между вычислительной эффективностью и точностью. Работа посвящена численному моделированию решетчатых метаматериалов, изготовленных с использованием аддитивных технологий. Объектом исследования является метаматериал с решетчатой структурой, предназначенный для применения в интрамедуллярных штифтах. Для исследования напряженно-деформированного состояния было построено пять конечно-элементных моделей, включающих балочные, оболочечные и твердотельные элементы в различных комбинациях и с различными граничными условиями. Для каждой модели выполнялась оценка сеточной сходимости эквивалентных напряжений по Мизесу в трех характерных точках изделия и результирующей реакции. Для верификации численных результатов использовались данные натурных испытаний решетчатых и сплошных образцов. Величина результирующей реакции в численном моделировании оказалась выше медианных данных натурных испытаний, но находилась в доверительном интервале. Модель, состоящая исключительно из твердотельных элементов, показала наибольшие размахи результирующей реакции при варьировании граничных условий. Наиболее точное совпадение с экспериментальными данными продемонстрировала смешанная модель, включающая балочные, оболочечные и твердотельные элементы. При этом баланс между точностью и временем вычислений обеспечивала модель, состоящая исключительно из балочных элементов. В расчетах использовался модуль упругости, полученный из натурных экспериментов сплошных образцов. Результаты работы подчеркивают необходимость учета специфики нерегулярных пористых структур при моделировании, включая макро- и мезопористость, остаточные напряжения и геометрические дефекты, возникающие при производстве или эксплуатации, а также важность верификации численных расчетов экспериментальными данными для повышения достоверности прогнозирования поведения имплантатов в клинических условиях.

This work is concerned with the mathematical and numerical analyses of the Navier–Stokes equations driven by threshold power law slip boundary condition. The tangential shear belongs to the sub-differential of a power law function, giving rise to quasi-variational inequalities. We construct a unique weak solution and formulate a mixed finite element approximation to the system using conforming finite element spaces. The unique solvability of the finite element approximation is investigated and an abstract convergence result is established. We derive a-priori error estimate by providing a bound on the error between the weak solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We introduce a Lagrange multiplier and propose a Uzawa-type iterative scheme to solve the nonlinear problem resulting from the finite element approximation. Finally, numerical results are provided to validate the theoretical findings.

Hybrid mixed discontinuous Galerkin finite element analysis of time-dependent convection–diffusion equations and its application in chemotaxis model

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

A robust Mixed Finite Element model for coupled Thermo-Hydro-Mechanical problems in unsaturated porous media

ABSTRACTWall shear stress (WSS) is a crucial hemodynamic quantity extensively studied in cardiovascular research, yet its numerical computation is not straightforward. This work compares WSS results obtained from two different finite element discretizations, quantifies the differences between continuous and discontinuous stresses, and introduces a modified variationally consistent method for WSS evaluation through the formulation of a boundary‐flux problem. Two benchmark problems are considered: a 2D Stokes flow on a unit square and a 3D Poiseuille flow through a cylindrical pipe. These are followed by investigations of steady‐state Navier–Stokes flow in two image‐based, patient‐specific aneurysms. The study focuses on P1/P1 stabilized and Taylor–Hood P2/P1 mixed finite elements for velocity and pressure. WSS is computed using either the proposed boundary‐flux method or as a projection of tangential traction onto first order Lagrange (P1), discontinuous Galerkin first order (DG‐1), or discontinuous Galerkin zero order (DG‐0) space. For the P1/P1 stabilized element, the boundary‐flux and P1 projection methods yielded equivalent results. With the P2/P1 element, the boundary‐flux evaluation demonstrated faster convergence in the Poiseuille flow example but showed increased sensitivity to pressure field inaccuracies in image‐based geometries compared to the projection method. Furthermore, a paradoxical degradation in WSS accuracy was observed when combining the P2/P1 element with fine boundary‐layer meshes on a cylindrical geometry, an effect attributed to inherent geometric approximation errors. In aneurysm geometries, the P2/P1 element exhibited superior robustness to mesh size when evaluating average WSS and low shear area (LSA), outperforming the P1/P1 stabilized element. Projecting discontinuous finite element functions into continuous spaces can introduce artifacts, such as the Gibbs phenomenon. Consequently, it is crucial to carefully select the finite element space for boundary stress calculations, not only in applications involving WSS computations for aneurysms.

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.

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

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

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