Energy Error Estimates Research Articles

A class of linearly implicit energy-preserving schemes for conservative systems

We consider a kind of differential equations y˙(t)=R(y(t))y(t)+f(y(t)) with energy conservation. Such conservative models appear for instance in quantum physics, engineering and molecular dynamics. A new class of energy-preserving schemes is constructed by the ideas of scalar auxiliary variable (SAV) and splitting, from which the nonlinearly implicit schemes have been improved to be linearly implicit. The energy conservation and error estimates are rigorously derived. Based on these results, it is shown that the new proposed schemes have unconditionally energy stability and can be implemented with a cost of solving a linearly implicit system. Numerical experiments are done to confirm these good features of the new schemes.

Numerical analysis of energy stable weak Galerkin schemes for the Cahn–Hilliard equation

In this paper, we construct and analyze the energy-stable weak Galerkin schemes for the Cahn–Hilliard equation in the mixed form. The energy stability depends on the newly defined discrete energy functional. We propose two robust weak Galerkin schemes with extra stabilizing terms in time: the first-order conditionally energy stable scheme of which the stability does not depend on spatial mesh size, and the second-order unconditionally energy stable scheme. Energy stabilities and error estimates for the fully discrete schemes are obtained. Numerical experiments are provided to validate the theoretical results. A classical phase transition model is presented to show the energy stable property of the schemes.

Optimal convergence analysis of the energy-preserving immersed weak Galerkin method for second-order hyperbolic interface problems in inhomogeneous media

This article reports our explorations for solving second-order hyperbolic interface problems by immersed weak Galerkin (IWG) method on interface independent meshes. The method presented here uses IWG functions for the discretization in spatial variable. The study includes the Newmark algorithm which has been used extensively in applications. The stability analysis based on the energy method is presented for semi-discrete and fully-discrete schemes under some conditions on parameters of the Newmark algorithm. In this work, we carried out a convergence analysis and obtained optimal a priori error estimates in both energy and L2 norms for the semi-discrete and fully-discrete schemes under piecewise H2 regularity assumption in space and some conditions on parameters of the Newmark algorithm. We demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(h2+τr(γ)), where r(γ)=1 if γ≠1/2, r(1/2)=2, h and τ are the mesh size and the time step, respectively. Numerical examples are provided to confirm theoretical findings and illustrate the efficiency of the method for standing and traveling waves.

Error analysis of the Crank–Nicolson SAV method for the Allen–Cahn equation on variable grids

The variable time-stepping technique is powerful in capturing the multi-scale behaviors (e.g., the solution changes rapidly in certain regions of time) for the Allen–Cahn equation. Based on the scalar auxiliary variable (SAV) approach and the Crank–Nicolson schemes, we establish the unconditional energy stability and error estimates rigorously for the Allen–Cahn equation on variable grids. A numerical experiment is performed to verify the theoretical results. To the best of our knowledge, this is the first topic on the convergence analysis for the SAV schemes on variable grids.

Unconditional Energy Dissipation and Error Estimates of the SAV Fourier Spectral Method for Nonlinear Fractional Generalized Wave Equation

In this paper, we consider a second-order scalar auxiliary variable Fourier spectral method to solve the nonlinear fractional generalized wave equation. Unconditional energy conservation or dissipation properties of the fully discrete scheme are first established. Next, we utilize the temporal-spatial error splitting argument to obtain unconditional optimal error estimate of the fully discrete scheme, which overcomes time-step restrictions caused by strongly nonlinear system, or the restrictions that the nonlinear term needs to satisfy the assumption of global Lipschitz condition in all previous works for fractional undamped or damped wave equations. Finally, some numerical experiments are presented to confirm our theoretical analysis.

A reduced-order finite element method based on proper orthogonal decomposition for the Allen-Cahn model

We focus on a reduced-order finite element (FE) method for the Allen-Cahn equation. Toward this end, we first derive a traditional FE (TFE) formulation for the Allen-Cahn equation and the energy stability and error estimates of the TFE solutions. Then we use the proper orthogonal decomposition (POD) technique to generate a set of POD bases from a few TFE solutions, establish a novel reduced-order FE (ROFE) formulation for the Allen-Cahn equation by the set of POD bases, and discuss the energy stability and error estimates of the ROFE solutions. Finally, we provide some numerical experiments to confirm the validity of the novel ROFE formulation.

A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems

<p style='text-indent:20px;'>We introduce an over-penalized weak Galerkin method for elliptic interface problems with non-homogeneous boundary conditions and discontinuous coefficients, where the method combines a weak Galerkin stabilizer with interior penalty terms. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements <inline-formula><tex-math id="M1">\begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{2}) $\end{document}</tex-math></inline-formula>. As an advantage of the method, elliptic interface problems with low regularity are approximated well. The over-penalized weak Galerkin method is based on weak functions whose edge part is double-valued on each interior edge sharing by two neighboring elements. Jumps between the edge parts are naturally used to define penalty terms. The over-penalized weak Galerkin method allows to use arbitrary meshes, even for low regularity solutions. These features make the new method more flexible and efficient for solving interface equations. Furthermore, <i>a priori</i> error estimates in energy and <inline-formula><tex-math id="M2">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula> norms are derived rigorously, and numerical results confirm the effectiveness of the method.

Stability analysis and error estimates of local discontinuous Galerkin methods with semi-implicit spectral deferred correction time-marching for the Allen–Cahn equation

This paper is concerned with the stability and error estimates of the local discontinuous Galerkin (LDG) method coupled with semi-implicit spectral deferred correction (SDC) time-marching up to third order accuracy for the Allen–Cahn equation. Since the SDC method is based on the first order convex splitting scheme, the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step, which increases the difficulty of the theoretical analysis. For the LDG discretizations coupled with the second and third order SDC methods, we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimensional spaces. At the same time, the iteration and integral involved in the semi-implicit SDC scheme also increase the difficulty of the theoretical analysis. Comparing to the Runge–Kutta type semi-implicit schemes which exclude the left-most endpoint, the SDC scheme in this paper includes the left-most endpoint as a quadrature node. This makes the test functions of the SDC scheme more complicated and the energy equations are more difficult to construct. We provide two different ideas to overcome the difficulty of the nonlinear terms. By choosing the test functions carefully, the energy stability and error estimates are obtained in the sense that the time step Δt only requires a positive upper bound and is independent of the mesh size h. Numerical examples are presented to illustrate our theoretical results.

A Hybrid High-Order Discretization Combined with Nitsche's Method for Contact and Tresca Friction in Small Strain Elasticity

We devise and analyze a Hybrid high-order (HHO) method to discretize unilateral and bilateral contact problems with Tresca friction in small strain elasticity. The nonlinear frictional contact conditions are enforced weakly by means of a consistent Nitsche's technique with symmetric, incomplete, and skew-symmetric variants. The present HHO-Nitsche method supports polyhedral meshes and delivers optimal energy-error estimates for smooth solutions under some minimal thresholds on the penalty parameters for all the symmetry variants. An explicit tracking of the dependency of the penalty parameters on the material coefficients is carried out to identify the robustness of the method in the incompressible limit, showing the more advantageous properties of the skew-symmetric variant. 2D and 3D numerical results including comparisons to benchmarks from the literature and to solutions obtained with an industrial software, as well as a prototype for an industrial application, illustrate the theoretical results and reveal that in practice the method behaves in a robust manner for all the symmetry variants in Nitsche's formulation.

A combined finite element method for parabolic equations posted in domains with rough boundaries

In this paper, we consider the solution of parabolic equations posted in domains with rough boundaries using the combined finite element procedure proposed in Xu et al. [A combined finite element method for elliptic problems posted in domains with rough boundaries, J. Comput. Appl. Math. 336 (2018), pp. 235–248] in space and backward-Euler scheme in time. We first prove error estimates for elliptic projection operator for both and by the dual argument which depend on the elliptic auxiliary problem, and then energy error estimates of semi-discrete and fully discrete schemes which have convergence rates about and , where s>0, respectively. Numerical results are provided for parabolic equations in domains with non-oscillating or oscillating boundaries to verify the theoretical findings.

Stabilization of high order cut finite element methods on surfaces

Abstract We develop and analyse a stabilization term for cut finite element approximations of an elliptic second-order partial differential equation on a surface embedded in ${\mathbb{R}}^d$. The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces, and provides control of the variation of the finite element solution on the active three-dimensional elements that intersect the surface. We show that the condition number of the stiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. The stabilization term works for linear as well as for higher-order elements and the derivation of its stabilizing properties is quite straightforward, which we illustrate by discussing the extension of the analysis to general $n$-dimensional smooth manifolds embedded in ${\mathbb{R}}^d$, with codimension $d-n$. We also state the properties of a general stabilization term that are sufficient to prove optimal scaling of the condition number and optimal error estimates in energy- and $L^2$-norm. We finally present numerical studies confirming our theoretical results.

Convergence of Adaptive Stochastic Galerkin FEM

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.

A Nonconforming Immersed Finite Element Method for Elliptic Interface Problems

A new immersed finite element (IFE) method is developed for second-order elliptic problems with discontinuous diffusion coefficient. The IFE space is constructed based on the rotated Q1 nonconforming finite elements with the integral-value degrees of freedom. The standard nonconforming Galerkin method is employed in this IFE method without any penalty stabilization term. Error estimates in energy and L2 norms are proved to be better than $O(h\sqrt{|\log h|})$ and $O(h^2|\log h|)$, respectively, where the logarithm factors reflect jump discontinuity. Numerical results are reported to confirm our analysis.

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, $Re$-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption $\nabla u \in L^1(0,T;L^\infty(\Omega))$ which is discussed in detail. In the sense of best practice, we review and establish pressure- and $Re$-semi-robust estimates for pointwise divergence-free $H^1$-conforming FEM (like Scott-Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free $H$(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.

An Advection-Robust Hybrid High-Order Method for the Oseen Problem

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer $k\ge 0$, the discrete velocity unknowns are vector-valued polynomials of total degree $\le k$ on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree $\le k$ on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree $\le(k+1)$, a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity-pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element $T$ of diameter $h_T$ contributes to the discretization error with an $\mathcal{O}(h_T^{k+1})$-term in the diffusion-dominated regime, an $\mathcal{O}(h_T^{k+\frac12})$-term in the advection-dominated regime, and scales with intermediate powers of $h_T$ in between. Numerical results complete the exposition.

Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes

We propose in this work a unified formulation of mixed and primal discretization methods on polyhedral meshes hinging on globally coupled degrees of freedom that are discontinuous polynomials on the mesh skeleton. To emphasize this feature, these methods are referred to here as discontinuous skeletal. As a starting point, we define two families of discretizations corresponding, respectively, to mixed and primal formulations of discontinuous skeletal methods. Each family is uniquely identified by prescribing three polynomial degrees defining the degrees of freedom, and a stabilization bilinear form which has to satisfy two properties of simple verification: stability and polynomial consistency. Several examples of methods available in the recent literature are shown to belong to either one of those families. We then prove new equivalence results that build a bridge between the two families of methods. Precisely, we show that for any mixed method there exists a corresponding equivalent primal method, and the converse is true provided that the gradients are approximated in suitable spaces. A unified convergence analysis is carried out delivering optimal error estimates in both energy- and L2-norms.

Interior energy error estimates for the weak Galerkin finite element method

Consider the Poisson equation in a polytopal domain $$\Omega \subset {\mathbb {R}}^d$$ ( $$d=2, 3$$ ) as the model problem. We study interior energy error estimates for the weak Galerkin finite element approximation to elliptic boundary value problems. In particular, we show that the interior error in the energy norm is bounded by three components: the best local approximation error, the error in negative norms, and the trace error on the element boundaries. This implies that the interior convergence rate can be polluted by solution singularities on the domain boundary, even when the solution is smooth in the interior region. Numerical results are reported to support the theoretical findings. To the best of our knowledge, this is the first local energy error analysis that applies to general meshes consisting of polytopal elements and hanging nodes.

Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection

In this paper, we propose a class of exponential time differencing (ETD) schemes for solving the epitaxial growth model without slope selection. A linear convex splitting is first applied to the energy functional of the model, and then Fourier collocation and ETD-based multistep approximations are used respectively for spatial discretization and time integration of the corresponding gradient flow equation. Energy stabilities and error estimates of the first and second order ETD schemes are rigorously established in the fully discrete sense. We also numerically demonstrate the accuracy of the proposed schemes and simulate the coarsening dynamics with small diffusion coefficients. The results show the logarithm law for the energy decay and the power laws for growth of the surface roughness and the mound width, which are consistent with the existing theories in the literature.

Electrostatic interactions in dissipative particle dynamics-Ewald-like formalism, error analysis, and pressure computation.

An accurate time evolution of charged species having exponentially smeared out charge density (Slater type charge distribution) in dissipative particle dynamic (DPD) simulations necessitates the optimal choice of the Ewald splitting parameter (α), charge smearing length (λ), and real space cutoff (c) when the Ewald summation or its variant such as particle-particle particle-mesh or particle-mesh Ewald is employed for long range electrostatics. The present article offers the error estimates in the electrostatic energy and the force as a function of α and β(1/λ) on account of spherical truncation c in real space. These error estimate formulae are validated by our DPD simulation results. We also give here an Ewald-like derivation for electrostatic energy and force for the Slater type charge density. A quick estimate of the electrostatic pressure without the use of the tedious expression which involves three dimensional Fourier transforms is also presented, and its range of validity is discussed. The basis for the proposed formula for pressure is the fact that the minimum-image truncation in many cases allows one to compute the thermodynamic quantities with reasonable accuracy.

OptimalL2velocity error estimate for a modified pressure-robust Crouzeix–Raviart Stokes element

Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix–Raviart element such that its velocity error becomes pressure-independent. The modification results in an O(h) consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the L norm of the pressure. However, though the optimal convergence of the velocity in the L norm was observed numerically, it appeared to be nontrivial to prove. In this contribution, this gap is closed. Moreover, the dependence of the energy error estimates on the discrete inf-sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numerical experiments in 2D and 3D illustrate the theoretical findings.