A-priori Error Estimates Research Articles

Polytopal discontinuous Galerkin discretization of brain multiphysics flow dynamics

A comprehensive mathematical model of the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain can be expressed as the coupling of a poromechanics system and Stokes' equations: the first describes fluids filtration through the cerebral tissue and the tissue's elastic response, while the latter models the flow of the CSF in the brain ventricles. This model describes the functioning of the brain's waste clearance mechanism, which has been recently discovered to play an essential role in the progress of neurodegenerative diseases. To model the interactions between different scales in the porous medium, we propose a physically consistent coupling between Multi-compartment Poroelasticity (MPE) equations and Stokes' equations. In this work, we introduce a numerical scheme for the discretization of such coupled MPE-Stokes system, employing a high-order discontinuous Galerkin method on polytopal grids to efficiently account for the geometric complexity of the domain. We analyze the stability and convergence of the space semidiscretized formulation, we prove a-priori error estimates, and we present a temporal discretization based on a combination of Newmark's β-method for the elastic wave equation and the θ-method for the other equations of the model. Numerical simulations carried out on test cases with manufactured solutions validate the theoretical error estimates. We also present numerical results on a two-dimensional slice of a patient-specific brain geometry reconstructed from diagnostic images, to test in practice the advantages of the proposed approach.

Numerical method for two-dimensional linearly elastic clamped plate model

ABSTRACT In this paper, we propose an efficient numerical method for the two-dimensional (2D) linearly elastic clamped plate model. Specifically, we choose the linear Lagrange element to discretize the first two components in the vector displacement space, as well as the Morley element to discretize the third component. Moreover, the existence of discrete solution, uniqueness and a-priori error estimate is analysed. Finally, numerical experiments are presented to verify theoretical results.

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

In this paper, with classic Legendre polynomials, a method of particular solutions (MPS, for short) is proposed to solve a kind of second-order differential equations with a variable coefficient on a unit interval. The particular solutions, satisfying the natural Dirichlet boundary conditions, are constructed with orthogonal Legendre polynomials for the variable coefficient case. Meanwhile, we investigate the a-priori error estimates of the MPS approximations. Two a-priori error estimations in H 1 - and L ∞ -norms are shown to depict the convergence order of numerical approximations, respectively. Some numerical examples and convergence rates are provided to validate the merits of our proposed meshless method.

A Numerical Study for a Flexible Euler-Bernoulli Beam with a Force Control in Velocity and a Moment Control in Rotating Velocity

In this paper, we numerically study a flexible Euler-Bernoulli beam with a force control in velocity and a moment control in rotating velocity. First, we show the existence and uniqueness of the weak solution using Faedo-Galerkin’s method with the intermediate spaces. Then, we use the finite elements method with the cubic Hermite polynomials for the approximation of (1.1)–(1.5) in space such that the semi-discrete scheme obtained is stable and convergent. In addition, an a-priori error estimate is obtained. Finally, we perform numerical simulations in order to validate this method.

Novel method of particular solutions for second-order variable coefficient differential equations on a square

In this work, a novel method of particular solutions (MPS, for short) is proposed with Legendre polynomials to solve second-order partial differential equations (PDEs) with a variable coefficient on a square. Specially, we prove the uniqueness of the solution for the given model, which is equipped with a sufficiently smooth variable coefficient excluding elliptic properties. Based on the characteristics of Legendre polynomials, particular solutions, which satisfy the natural boundary condition, are constructed for the variable coefficient within the PDEs. Meanwhile we investigate the a-priori error estimates in $ H^1 $-norm and $ L^2 $-norm of the MPS approximations, which depict the asymptotic super-exponential convergence rates of the numerical approximations for sufficiently smooth solutions. Some numerical examples and convergence rates are listed to depict the highlights of this proposed meshless method.

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

On Mathematical and Numerical Modelling of Multiphysics Wave Propagation with Polytopal Discontinuous Galerkin Methods: a Review

In this work we review discontinuous Galerkin finite element methods on polytopal grids (PolydG) for the numerical simulation of multiphysics wave propagation phenomena in heterogeneous media. In particular, we address wave phenomena in elastic, poro-elastic, and poro-elasto-acoustic materials. Wave propagation is modeled by using either the elastodynamics equation in the elastic domain, the acoustics equations in the acoustic domain and the low-frequency Biot’s equations in the poro-elastic one. The coupling between different models is realized by means of (physically consistent) transmission conditions, weakly imposed at the interface between the subdomains. For all models configuration, we introduce and analyse the PolydG semi-discrete formulation, which is then coupled with suitable time marching schemes. For the semi-discrete problem, we present the stability analysis and derive a-priori error estimates in a suitable energy norm. A wide set of two-dimensional verification tests with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also shown to demonstrate the capability of the proposed methods.

Optimal error bound for immersed weak Galerkin finite element method for elliptic interface problems

In Mu and Zhang (2019), an immersed weak Galerkin finite element method (IWG-FEM) is developed for solving elliptic interface problems and it is proved that this method has optimal a-priori error estimate in an energy norm under artificial smoothness assumption on the solution. In this study, we prove that IWG-FEM converges optimally in energy norm under natural smoothness assumption on solution. Furthermore, we show that IWG-FEM converges optimally in the L2 norm which did not present in Mu and Zhang (2019) because of the artificial H3 smoothness requirement. A series of numerical experiments are conducted and reported to verify the theoretical finding.

An implicit\u2013explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

We construct and analyze a second-order implicit–explicit (IMEX) scheme for the time integration of semilinear second-order wave equations. The scheme treats the stiff linear part of the problem implicitly and the nonlinear part explicitly. This makes the scheme unconditionally stable and at the same time very efficient, since it only requires the solution of one linear system of equations per time step. For the combination of the IMEX scheme with a general, abstract, nonconforming space discretization we prove a full discretization error bound. We then apply the method to a nonconforming finite element discretization of an acoustic wave equation with a kinetic boundary condition. This yields a fully discrete scheme and a corresponding a-priori error estimate.

A Finite Element Method for a Fourth Order Surface Equation With Application to the Onset of Cell Blebbing

A variational problem for a fourth order parabolic surface partial differential equation is discussed. It contains nonlinear lower order terms, on which we only make abstract assumptions, and which need to be defined for specified problems. We derive a semi-discrete scheme based on the surface finite element method, show a-priori error estimates, and use the analytical results to prove well-posedness. Furthermore, we present a computational framework where specific problems can be conveniently implemented and, later on, altered with relative ease. It uses a domain specific language implemented in Python. The high level program control can also be done within the Python scripting environment. The computationally expensive step of evolving the solution over time is carried out by binding to an efficient C++ software back-end. The study is motivated by cell blebbing, which can be instrumental for cell migration. Starting with a force balance for the cell membrane, we derive a continuum model for some mechanical and geometrical aspects of the onset of blebbing in a form that fits into the abstract framework. It is flexible in that it allows for amending force contributions related to membrane tension or the presence of linker molecules between membrane and cell cortex. Cell membrane geometries given in terms of a parametrisation or obtained from image data can be accounted for by the software. The use of a domain specific language to describe the model makes is straightforward to add additional effects such as reaction-diffusion equations modelling some biochemistry on the cell membrane. Some numerical simulations illustrate the approach.

An $h$-$p$ version of the Chebyshev spectral collocation method for Volterra integro-differential equations with vanishing delays

We propose a spectral collocation method based on the Chebyshev–Gauss–Lobatto points for linear Volterra integro-differential equations with vanishing delays. We derive an h-p version a-priori error estimate in the H1-norm that is completely explicit in the local time steps and the local polynomial degrees. Numerical experiments are provided to confirm the theoretical results.

Galerkin method with trigonometric basis on stable numerical differentiation

This paper considers the p (p=1,2,3) order numerical differentiation on function y in (0, 2π). They are transformed into corresponding Fredholm integral equation of the first kind. Computational schemes with analytic solution formulas are designed using Galerkin method on trigonometric basis. Convergence and divergence are all analysed in Corollaries 5.1, 5.2, and a-priori error estimate is uniformly obtained in Theorem 6.1, 7.1, 7.2. Therefore, the algorithm achieves the optimal convergence rate O(δ2μ2μ+1)(μ=12or1) with periodic Sobolev source condition of order 2µp. Besides, we indicate a noise-independent a-priori parameter choice when the function y possesses the form of∑k=0p−1aktk+∑k=1N1bkcoskt+∑k=1N2cksinkt,bN1,cN2≠0,In particular, in numerical differentiations for functions above, good filtering effect (error approaches 0) is displayed with corresponding parameter choice. In addition, several numerical examples are given to show that even derivatives with discontinuity can be recovered well.

Discontinuous Galerkin methods for short pulse type equations via hodograph transformations

In the present paper, we consider the discontinuous Galerkin (DG) methods for solving short pulse (SP) type equations. The short pulse equation has been shown to be completely integrable, which admits the loop-soliton, cuspon-soliton solutions as well as smooth-soliton solutions. Through hodograph transformations, these nonclassical solutions can be profiled as the smooth solutions of the coupled dispersionless (CD) system or the sine-Gordon equation. Therefore, DG methods can be developed for the CD system or the sine-Gordon equation to simulate the loop-soliton or cuspon-soliton solutions of the SP equation. The conservativeness or dissipation of the Hamiltonian or momentum for the semi-discrete DG schemes can be proved. Also we modify the above DG schemes and obtain an integration DG scheme. Theoretically the a-priori error estimates have been provided for the momentum conserved DG scheme and the integration DG scheme. We also propose the DG scheme and the integration DG scheme for the sine-Gordon equation, in case the SP equation can not be transformed to the CD system. All these DG schemes can be applied to the generalized or modified SP type equations. Numerical experiments are provided to illustrate the optimal order of accuracy and capability of these DG schemes.

Stability analysis and best approximation error estimates of discontinuous time-stepping schemes for the Allen–Cahn equation

Fully-discrete approximations of the Allen–Cahn equation are considered. In particular, we consider schemes of arbitrary order based on a discontinuous Galerkin (in time) approach combined with standard conforming finite elements (in space). We prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. We also prove best approximation a-priori error estimates, with constants depending polynomially upon (1/ε) by circumventing Gronwall Lemma arguments. The key feature of our approach is a carefully constructed duality argument, combined with a boot-strap technique.

A lowest-order weak Galerkin method for linear elasticity

The lowest-order weak Galerkin (WG) method is considered for linear elasticity based on the displacement formulation. The new method approximates the displacement using piecewise constant vector functions both inside and on the boundary of each mesh element and its bilinear form does not require a stabilization term for the existence and uniqueness of the solution. A-priori error estimates of optimal order in the discrete H 1 - and L 2 -norms for the displacement are proved when the solution is smooth. The error estimates are independent of the Lamé constant λ , thus the performance of the new method does not deteriorate as the elastic material becomes incompressible. Further, a simple post-processing technique to obtain a numerical approximation of the stress is presented. A careful error analysis reveals that the L 2 -norm error in the stress is also optimal and independent of λ . Several numerical experiments confirm the locking-free property and optimal convergence rates of the new method.

Local projection stabilization for convection–diffusion–reaction equations on surfaces

The numerical solution of convection–diffusion–reaction equations in two and three dimensional domains Ω is thoroughly studied and well understood. Stabilized finite element methods have been developed to handle boundary or interior layers and to localize and suppress unphysical oscillations. Much less is known on convection–diffusion–reaction equations on surfaces Γ=∂Ω. We propose a fitted finite element local projection stabilization (LPS) for convection–diffusion–reaction equations on surfaces based on a linear surface approximation and continuous, piecewise linear finite elements. Unique solvability of the continuous and discrete problem is established. A-priori error estimates show the O(h3∕2) convergence in mesh-dependent norms for smooth solutions in the convection-dominated case. The associated bilinear form satisfies an inf–sup condition in a norm stronger than the standard LPS-norm leading to improved stability properties. Numerical test examples demonstrate the properties and the potential of the proposed method.

An analysis of the TDNNS method using natural norms

The tangential-displacement normal-normal-stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing. We present a sound mathematical theory of the infinite dimensional problem using the space {{mathbf {H}}}(mathbf {curl}) for the displacement. We define the space {underline{{mathbf {H}}}}({text {div}},mathbf {{div}}) for the stresses and provide trace operators for the normal-normal stress. Moreover, the finite element problem is shown to be stable with respect to the {{mathbf {H}}}(mathbf {curl}) and a discrete {underline{{mathbf {H}}}}({text {div}},mathbf {{div}}) norm. A-priori error estimates of optimal order with respect to these norms are obtained. Beside providing a new analysis for the elasticity equation, the numerical techniques developed in this paper are a foundation for more complex models from structural mechanics such as Reissner Mindlin plate equations, see Pechstein and Schöberl (Numerische Mathematik 137(3):713–740, 2017).

Isogeometric collocation for Kirchhoff–Love plates and shells

With the emergence of isogeometric analysis (IGA), the Galerkin rotation-free discretization of Kirchhoff–Love shells is facilitated, enabling more efficient thin shell structural analysis. High-order shape functions used in IGA also allow the collocation of partial differential equations, avoiding the time-consuming numerical integration of the Galerkin technique. The goal of the present work is to apply this method to NURBS-based isogeometric Kirchhoff–Love plates and shells, under the assumption of small deformations.Since Kirchhoff–Love plate theory yields a fourth-order formulation, two boundary conditions are required at each location on the contour, generating some conflicts at the corners where there are more equations than needed. To remedy this overdetermination, we provide priority and averaging rules that cover all the possible combinations of adjacent edge boundary conditions (i.e. the clamped, simply-supported, symmetric and free supports). Greville and alternative superconvergent points are used for NURBS basis of even and odd degrees, respectively. For square, circular, and annular flat plates, convergence orders are found to be in agreement with a-priori error estimates. The proposed isogeometric collocation method is then validated and benchmarked against a Galerkin implementation by studying a set of problems involving Kirchhoff–Love shells.

Dissipative Numerical Method for a Flexible Euler-Bernoulli Beam with a Force Control in Rotation and Velocity Rotation

In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation. We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case. We prove firstly the existence and uniqueness of the weak solution. Then, we discretize the system in two steps: in the first step, a semi-discrete scheme is obtained for discretization in space and, in the second step, a fully-discrete scheme is obtained for discretization in time by the Crank-Nicolson scheme. At each step of the discretization, the a-priori error estimates are obtained.

Convergence estimates for a plane elasticity problem solved by the Galerkin boundary integral formulation with NURBS

A classical mixed boundary-value problem of linear elasticity in two dimensions in the Galerkin boundary integral formulation is considered. We prove a-priori error estimates of the solution to this problem by the isogeometric adaptive method using NURBS. The estimates include approximation of the boundary of the domain.