Standard Continuous Finite Element Research Articles

Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps

In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements with a stabilization consisting of a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that for the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally the L2-norm error converges with the expected order O(h^{k+frac 12}), if the exact solution is locally smooth. We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if underresolved features have been present in the solution at some time during the evolution.

Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems

The algebraic flux correction (AFC) schemes presented in this work constrain a standard continuous finite element discretization of a nonlinear hyperbolic problem to satisfy relevant maximum principles and entropy stability conditions. The desired properties are enforced by applying a limiter to antidiffusive fluxes that represent the difference between the high-order baseline scheme and a property-preserving approximation of Lax–Friedrichs type. In the first step of the limiting procedure, the given target fluxes are adjusted in a way that guarantees preservation of local and/or global bounds. In the second step, additional limiting is performed, if necessary, to ensure the validity of fully discrete and/or semi-discrete entropy inequalities. The limiter-based entropy fixes considered in this work are applicable to finite element discretizations of scalar hyperbolic equations and systems alike. The underlying inequality constraints are formulated using Tadmor’s entropy stability theory.The proposed limiters impose entropy-conservative or entropy-dissipative bounds on the rate of entropy production by antidiffusive fluxes and Runge–Kutta (RK) time discretizations. Two versions of the fully discrete entropy fix are developed for this purpose. The first one incorporates temporal entropy production into the flux constraints, which makes them more restrictive and dependent on the time step. The second algorithm interprets the final stage of a high-order AFC-RK method as a constrained antidiffusive correction of an implicit low-order scheme (algebraic Lax–Friedrichs in space + backward Euler in time). In this case, iterative flux correction is required, but the inequality constraints are less restrictive and limiting can be performed using algorithms developed for the semi-discrete problem. To motivate the use of limiter-based entropy fixes, we prove a finite element version of the Lax–Wendroff theorem and perform numerical studies for standard test problems. In our numerical experiments, entropy-dissipative schemes converge to correct weak solutions of scalar conservation laws, of the Euler equations, and of the shallow water equations.

A subcell-enriched Galerkin method for advection problems

In this work, we introduce a generalization of the enriched Galerkin (EG) method. The key feature of our scheme is an adaptive two-mesh approach that, in addition to the standard enrichment of a conforming finite element discretization via discontinuous degrees of freedom, allows to subdivide selected (e.g. troubled) mesh cells in a non-conforming fashion and to use further discontinuous enrichment on this finer submesh. We prove stability and sharp a priori error estimates for a linear advection equation by using a specially tailored projection and conducting some parts of a standard convergence analysis for both meshes. By allowing an arbitrary degree of enrichment on both, the coarse and the fine mesh (also including the case of no enrichment), our analysis technique is very general in the sense that our results cover the range from the standard continuous finite element method to the standard discontinuous Galerkin (DG) method with (or without) local subcell enrichment. Numerical experiments confirm our analytical results and indicate good robustness of the proposed method.

Solving Galbrun's Equation with a Discontinuous galerkin Finite Element Method

Over many years, scientists and engineers have developed a broad variety of mathematical formulations to investigate the propagation and interactions with flow of flow-induced noise in early-stage of product design and development. Beside established theories such as the linearized Euler equations (LEE), the linearized Navier–Stokes equations (LNSE) and the acoustic perturbation equations (APE) which are described in an Eulerian framework, Galbrun utilized a mixed Lagrange–Eulerian framework to reduce the number of unknowns by representing perturbations by means of particle displacement only. Despite the advantages of fewer degrees of freedom and the reduced effort to solve the system equations, a computational approach using standard continuous finite element methods (FEM) suff ers from instabilities called spurious modes that pollute the solution. In this work, the authors employ a discontinuous Galerkin approach to overcome the difficulties related to spurious modes while solving Galbrun's equation in a mixed and pure displacement based formulation. The results achieved with the proposed approach are compared with results from previous attempts to solve Galbrun's equation. The numerical determination of acoustic modes and the identification of vortical modes is discussed. Furthermore, case studies for a lined-duct and an annulus supporting a rotating shear-flow are investigated.

Primal Dual Mixed Finite Element Methods for Indefinite Advection-Diffusion Equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.

An Efficient Finite Element Approach for Modeling Fibrotic Clefts in the Heart

Advanced medical imaging technologies provide a wealth of information on cardiac anatomy and structure at a paracellular resolution, allowing to identify microstructural discontinuities which disrupt the intracellular matrix. Current state-of-the-art computer models built upon such datasets account for increasingly finer anatomical details, however, structural discontinuities at the paracellular level are typically discarded in the model generation process, owing to the significant costs which incur when using high resolutions for explicit representation. In this study, a novel discontinuous finite element (dFE) approach for discretizing the bidomain equations is presented, which accounts for fine-scale structures in a computer model without the need to increase spatial resolution. In the dFE method, this is achieved by imposing infinitely thin lines of electrical insulation along edges of finite elements which approximate the geometry of discontinuities in the intracellular matrix. Simulation results demonstrate that the dFE approach accounts for effects induced by microscopic size scale discontinuities, such as the formation of microscopic virtual electrodes, with vast computational savings as compared to high resolution continuous finite element models. Moreover, the method can be implemented in any standard continuous finite element code with minor effort.

Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation

In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process {u(t)}t∈[0,T ] satisfying a stochastic linear evolution equation with a positive-type memory term driven by an additive Gaussian noise. The equation can be written in an abstract form as du+ (∫ t 0 b(t− s)Au(s) ds ) dt = dW , t ∈ (0, T ]; u(0) = u0 ∈ H, where W is a Q-Wiener process on H = L(D) and where the main example of b we consider is given by b(t) = tβ−1/Γ (β), 0 0 such that A−α has finite trace and that Q is bounded from H into D(A) for some real κ with α− 1 β+1 < κ ≤ α. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter ∆t = T/n), and a standard continuous finite element method in space (parameter h). Let un,h be the discrete solution at T = n∆t. We show that ( E‖un,h − u(T )‖ )1/2 = O(h +∆t), for any γ < (1− (β + 1)(α− κ))/2 and ν ≤ 1 β+1 − α+ κ. M. Kovacs Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, 9054, New Zealand. E-mail: mkovacs@maths.otago.ac.nz J. Printems Laboratoire d’Analyse et de Mathematiques Appliquees CNRS UMR 8050, 61, avenue du General de Gaulle, Universite Paris–Est, 94010 Creteil, France. E-mail: printems@u-pec.fr 2 Mihaly Kovacs, Jacques Printems

Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term

In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract Itô form asdX(t)+(∫0tb(t−s)AX(s)ds)dt=dWQ(t),t∈(0,T];X(0)=X0∈H, where WQ is a Q-Wiener process on the Hilbert space H and where the time kernel b is the locally integrable potential tρ−2, ρ∈(1,2), or slightly more general. The operator A is unbounded, linear, self-adjoint, and positive on H. Our main assumption concerning the noise term is that A(ν−1/ρ)/2Q1/2 is a Hilbert–Schmidt operator on H for some ν∈[0,1/ρ]. The numerical approximation is achieved via a standard continuous finite element method in space (parameter h) and an implicit Euler scheme and a Laplace convolution quadrature in time (parameter Δt=T/N). We show that for φ:H→R twice continuously differentiable test function with bounded second derivative,|Eφ(XhN)−Eφ(X(T))|⩽Cln(Th2/ρ+Δt)(Δtρν+h2ν), for any 0⩽ν⩽1/ρ. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.

An Asymptotic Preserving Method for Linear Systems of Balance Laws Based on Galerkin’s Method

We apply the concept of Asymptotic Preserving (AP) schemes to the linearized p-system and discretize the resulting elliptic equation using standard continuous Finite Elements instead of Finite Differences. The fully discrete method is analyzed with respect to consistency, and we compare it numerically with more traditional methods such as Implicit Euler's method. Numerical results indicate that the AP method is indeed superior to more traditional methods.

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation.

A cohesive crack propagation model: Mathematical theory and numerical solution

We investigate the propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. As we take cohesive effects along the crack path into account and impose a non-penetration condition, inequalities appear in the constitutive equations describing the elastic behavior of a domain with crack. In contrast to existing approaches, we consider cohesive effects arising from crack opening in normal as well as in tangential direction. We establish a constrained energy minimization problem and show that the solution of this problem satisfies the set of constitutive equations. In order to solve the energy minimization problem numerically, we apply a finite element discretization using a combination of standard continuous finite elements with so-called cohesive elements. A particular strength of our method is that the crack path is a result of the minimization process. We conclude the article by numerical experiments and compare our results to results given in the literature.

Towards optimization of crack resistance of composite materials by adjustment of fiber shapes

We investigate the evolution and propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. In addition to the classical variational formulation, where the standard potential energy is minimized over the cracked domain under physical conditions characterizing the behavior of the material close to the crack (e.g. non-penetration conditions), we include a ‘cohesive traction term’ in the energy expression. In this way we obtain a mathematically concise set of partial differential equations with non-linear boundary conditions at the crack interfaces. We perform a finite element discretization using a combination of standard continuous finite elements and so-called cohesive elements. During the simulation process cohesive elements are adaptively inserted at positions where a certain stress bound is exceeded. In our numerical studies we consider domains consisting of a matrix material with fiber inclusions. Beyond pure crack path simulation, our ultimate goal is to determine an optimal shape of the fibers resulting in a crack path that releases for a given load scenario as much energy as possible without destroying the specimen completely. We develop a corresponding optimization model and propose a solution algorithm for the same. The article is concluded by numerical results.

A discontinuous Galerkin method for the wave equation

We present a new discontinuous Galerkin method for solving the second-order wave equation using the standard continuous finite element method in space and a discontinuous method in time directly applied to second-order ode systems. We prove several optimal a priori error estimates in space–time norms for this new method and show that it can be more efficient than existing methods. We also write the leading term of the local discretization error in terms of Lobatto polynomials in space and Jacobi polynomials in time which leads to superconvergence points on each space–time cell. We discuss how to apply our results to construct efficient and asymptotically exact a posteriori estimates for space–time discretization errors. Numerical results are in agreement with theory.

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell's equations. An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation and a more general family of schemes applied to the corresponding first order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher order approximation in space

On a coupled finite element-finite volume method for convection-diffusion problems

In this paper we present and analyse a coupled finite element-finite volume method for the numerical approximation of singularly perturbed convection-diffusion problems. The idea is to couple a discretization for the convective term, based on the finite volume (node-centred) method, and a standard continuous finite element approximation of the diffusive term. Such a method preserves conservation, fulfils consistency and enhances stability.

Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in 𝑅³

This paper is devoted to the steady state, incompressible Navier-Stokes equations with nonstandard boundary conditions of the form u ⋅ n = 0 {\mathbf {u}} \cdot {\mathbf {n}} = 0 , c u r l u × n = 0 \mathbf {curl}\;{\mathbf {u}} \times {\mathbf {n}} = {\mathbf {0}} , either on the entire boundary or mixed with the standard boundary condition u = 0 {\mathbf {u}} = {\mathbf {0}} on part of the boundary. The problem is expressed in terms of vector potential, vorticity and pressure. The vorticity and vector potential are approximated with curl-conforming finite elements and the pressure with standard continuous finite elements. The error estimates yield nearly optimal results for the purely nonstandard problem.