Nonconforming Finite Element Research Articles

A unified error analysis for nonlinear wave-type equations with application to acoustic boundary conditions

In this work we present a unified error analysis for abstract space discretizations of wave-type equations with nonlinear quasi-monotone operators. This yields an error bound in terms of discretization and interpolation errors that can be applied to various equations and space discretizations fitting in the abstract setting. We use the unified error analysis to prove novel convergence rates for a non-conforming finite element space discretization of wave equations with nonlinear acoustic boundary conditions and illustrate the error bound by a numerical experiment.

Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

AbstractIn this article, we propose and analyze several numerical methods for the nonlinear delay reaction–diffusion system with smooth and nonsmooth solutions, by using Quasi‐Wilson nonconforming finite element methods in space and finite difference methods (including uniform and nonuniform L1 and L2‐1σ schemes) in time. The optimal convergence results in the senses of L2‐norm and broken H1‐norm, and H1‐norm superclose results are derived by virtue of two types of fractional Grönwall inequalities. Then, the interpolation postprocessing technique is used to establish the superconvergence results. Moreover, to improve computational efficiency, fast algorithms by using sum‐of‐exponential technique are built for above proposed numerical schemes. Finally, we present some numerical experiments to confirm the theoretical correctness and show the effectiveness of the fast algorithms.

Nonconforming finite element formulation for the simulation of impedance cardiography

AbstractWe present a nonconforming Finite Element (FE) formulation for the electro‐quasistatic case. The formulation allows a grid generation for each sub‐domain individually. At the nonconforming interfaces, we apply a Nitsche‐type mortaring, which can also be interpreted as a DG‐IP (Discontinuous Galerkin‐Internal Penalty) formulation. Thereby, the continuity of the electric potential and the electric flux in the discrete setting is guaranteed. The system matrix remains symmetric, and the necessary penalty factor can be chosen in a wide range without deteriorating the numerical solution. The application to impedance cardiography demonstrates the practical applicability, especially its advantage toward simplification of the grid generation.

A new energy-stable nonconforming finite element method for Sobolev equation with Burgers’ type nonlinearity

We propose an energy-stable finite element method (FEM) of the Sobolev equation with Burgers’ type nonlinearity by implicit Euler method in time and nonconforming EQ1rot element in space. A stabilized term is innovatively added in this work to preserve the energy dissipation of the numerical scheme. Then we directly obtain the prior estimates of the numerical solution, with which the unique solvability of the fully-discrete scheme is derived. Subsequently, a novel strategy is utilized to achieve the superclose estimate of order O(h2+τ) in broken H1-norm. Using the interpolated postprocessing technique, the global superconvergence result is obtained. At last, numerical example is given to confirm the theoretical analysis. Here, h is the spatial parameter, and τ is the time step.

A subspace of linear nonconforming finite element for nearly incompressible elasticity and Stokes flow

Abstract The linear nonconforming finite element, combined with constant finite element for pressure, is stable for the Stokes problem. But it does not satisfy the discrete Korn inequality. The linear conforming finite element satisfies the discrete Korn inequality, but is not stable for the Stokes problem and fails for the nearly incompressible elasticity problems. We enrich the linear conforming finite element by some nonconforming P1 bubbles, i.e., select a subspace of the linear nonconforming finite element space, so that the resulting linear nonconforming element is both stable and conforming enough to satisfy the Korn inequality, on HTC-type triangular and tetrahedral grids. Numerical tests in 2D and 3D are presented, confirming the analysis.

Robust Monolithic Solvers for the Stokes--Darcy Problem with the Darcy Equation in Primal Form

We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and non-conforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.

Optimal H1-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

In this paper, based on the L2-1σ scheme and nonconforming EQ1rot finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and detailed analysis of the equations with an initial singularity is described on anisotropic meshes. The fully discrete scheme is shown to be unconditionally stable, and optimal second-order accuracy for convergence and superconvergence can be achieved in both time and space directions. Finally, the obtained numerical results are compared with the theoretical analysis, which verifies the accuracy of the proposed method.

Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations

Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations

Optimal Convergence Analysis of Two-Level Nonconforming Finite Element Iterative Methods for 2D/3D MHD Equations.

Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods.

A locking-free nonconforming FEM for optimal control problems governed by linear elasticity equations

In this paper, we investigate a locking-free nonconforming finite element method (FEM) for linear elasticity optimal control problems (OCPs) via the variational control discretization approach. Due to the fact that the finite element interpolation operator is identical to the Ritz projection, optimal error estimates are established for the control, state and adjoint state variables, respectively. Numerical experiments are also provided to validate the theoretical results.

Convergence of adaptive nonconforming finite element method for Stokes optimal control problems

This paper aims at proving the convergence and quasi-optimality of an adaptive nonconforming finite element method for Stokes distributed control problems with pointwise control constraints. Nonconforming P1/P0 pair (Crouzeix–Raviart elements) and variational discretization are used to approximate the state equation and the control variable, respectively. A posteriori error estimates with upper and lower bounds are first derived for the state and adjoint variables. Then we prove the contraction property for the sum of the energy error of the state and adjoint state and the scaled error estimator on two consecutive adaptive meshes. The resulting linear convergence is finally used to show the quasi-optimal convergence rate of the adaptive algorithm. Additionally, some numerical results are provided to support our theoretical analysis.

3D meso-scale investigation of ultra high performance fibre reinforced concrete (UHPFRC) using cohesive crack model and Weibull random field

Ultra high performance fibre reinforced concrete (UHPFRC) is composed of fibres, mortar, fibre-mortar interfaces, and pores randomly distributed in space, and thus has heterogeneous mechanical properties and complicated failure mechanisms at micro/meso-scales. This study aims to develop a highly effective 3D finite element modelling approach to simulate both the mortar heterogeneity and the complicated failure mechanisms in UHPFRC materials and structural members. In this approach, Weibull random fields are used to characterize random properties (such as moduli, strength and fracture energy) of the mortar caused by defects such as pores and microcracks; pre-inserted cohesive interface elements with softening constitutive laws are applied to model multi-cracking behaviour; specific probability density distributions are adopted to describe the random orientation of fibres in the mortar, and orientation-dependent bond-slip relations are developed to model fibre-mortar interactions. The fibres are rigidly embedded in the mortar so that non-conforming finite element meshes can be used conveniently. The modelling approach has been validated by extensive simulations of single fibre pull-out tests and direct tensile tests of UHPFRC specimens with thousands of fibres in agreement with experimental data. It is found that the modelling approach is highly efficient in capturing the fibre crack-bridging effect, simulating the damage and fracture initiation and evolution, and quantifying the load-carrying capacities with statistical analyses considering fibre orientation and mortar heterogeneity.

Two-level Newton iterative method based on nonconforming finite element discretization for 2D/3D stationary MHD equations

In this paper, two-level Newton iterative method based on nonconforming finite element discretization is presented for solving 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix–Raviart type element for the velocity, and the conforming finite element for the magnetic field and pressure. The main idea of the proposed method is to solve MHD system by m Newton iterations on a coarse mesh, once correction by Stokes iteration on a fine mesh. The proposed method can save more computational time than one level method on the fine mesh with the same convergence rate. Moreover, the technical analysis of stability and optimal error estimates for two-level Newton iterative method are given. Finally, the applicability and efficiency of our proposed algorithm are illustrated by several numerical experiments.

Efficient numerical approach for solving the diffusion equation with variable coefficients

We propose and investigate an N-layered diffusion model with piecewise constant coefficients for approximating the exact solution of the diffusion equation with depth-dependent material coefficients in the spatial frequency domain. It is shown that this numerical approach, which is quite easy in view of the implementation, exhibits a convergence rate of O ( N − 2 ) in the discrete L ∞ -norm and hence provides an interesting alternative to frequently used numerical approaches such as finite element or finite difference methods. For comparison purposes, we take into account the classical finite element approach under the use of continuous linear basis functions as well as a recently reported nonconforming finite element discretization - the weak Gakerkin method - that employs discontinuous functions in form of locally defined polynomials.

A Two-Level Nonconforming Rotated Quadrilateral Finite Element Method for the Stationary Navier–Stokes Equations

In this paper, we propose a two-level nonconforming rotated finite element (TNRFE) method for solving the Navier–Stokes equations. A new nonconforming rotated finite element (NRFE) method was proposed by Douglas added by conforming bubbles to velocity and discontinuous piecewise constant to the pressure on quadrilateral elements possessing favorable stability properties. The TNRFE method involves solving a small Navier–Stokes problem on a coarse mesh with mesh size H and a large linearized Navier–Stokes problem on a fine mesh with mesh size h by the NRFE method. If we choose h = O H 2 , the TNRFE method gives the convergence rate of the same order as that of the NRFE method. Compared with the NRFE method, the TNRFE method can save a large amount of CPU time. In this paper, the stability of the approximate solutions and the error estimates are proved. Finally, the numerical experiments are given, and results indicate that the method is practicable and effective.

A Sequentially- Hybridized Locally Conservative Non-conforming Finite Element Scheme for Two-phase Flow Simulation through Heterogeneous Porous Media

Accurate two-phase flow modeling has a crucial role in detecting pollution in aquifers and/or in the simulating immiscible flow through porous media. In this research, a new hybrid numerical method based on the lower order non-conforming finite element method (NCFEM) and interior penalty discontinuous Galerkin (DG) method is proposed to simulate two-phase incompressible flow in heterogeneous porous media. The pressure and transport saturation equations are discretized by using the non-conforming Crouzeix-Raviart (CR) finite element and the symmetric weighed interior penalty discontinuous Galerkin (SWIPM). As a result of determining the degrees of freedom at the midpoint of neighboring element edges, a consistent set of pressure and velocity fields is obtained via the NCFEM. An H(div) projection based on the Raviart-Thomas (RT) element is implemented to improve the results resolution and preserve the continuity of the normal component of the velocity field. Besides, the spurious numerical oscillations of the saturation solution at the end of each time step are removed by using a novel vertex-based slope limiter, named the modified Chavent-Jaffre limiter. The accuracy and performance of the proposed numerical method are assessed by solving the Buckley-Leverett and McWhorter-Sunada benchmark problems, along with two test problems associated with highly heterogeneous porous media. The numerical results indicate that the proposed scheme has a remarkable potential to accurately capture the shock front and sharp interface zone of immiscible phases in the heterogeneous aquifers and other porous media.

A virtual class of non-conforming finite elements and its applications to Poisson's equation

A virtual class of non-conforming finite elements and its applications to Poisson's equation

Comparative Study of Space Iteration Methods Based on Nonconforming Finite Element for Stationary Navier-Stokes Equations

Comparative Study of Space Iteration Methods Based on Nonconforming Finite Element for Stationary Navier-Stokes Equations

A virtual class of non-conforming finite elements and its applications to Poisson's equation

In this paper, we develop and analyse the virtual class of non-conforming finite element inspired from the classical Crouzeix-Raviart element in two and three-dimensions. We focus on Poisson's equation for theoretical and numerical results. We show that the discrete problem is stable and the a priori error estimates are optimal. Two numerical tests are presented to demonstrate the theoretical results.

A Posteriori Error Estimates for a Nonconforming Finite Element Discretization of the Stokes–Biot System

This paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the problem defining the interaction between a free fluid and poroelastic structure. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. The approach utilizes the same nonconforming Crouzeix–Raviart element discretization on the entire domain. For this discretization, we derive a residual indicator based on the jumps of the normal derivative of the nonconforming approximation. Lower and upper bounds form the main results with minimal assumptions on the mesh.