Spectral Element Approximation Research Articles

Positivity-preserving discontinuous spectral element methods for compressible multi-species flows

We introduce a novel positivity-preserving numerical stabilisation approach for high-order discontinuous spectral element approximations of compressible multi-species flows. The underlying stabilisation method is the adaptive entropy filtering approach (Dzanic and Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative formulation of the multi-species flow equations. We show that the straightforward enforcement of entropy constraints in the filter yields poor results around species interfaces and propose an adaptive switch for the entropy bounds based on the convergence properties of the pressure field which drastically improves its performance for multi-species flows. The proposed approach is shown in a variety of numerical experiments applied to the multi-species Euler and Navier–Stokes equations computed on unstructured grids, ranging from shock-fluid interaction problems to three-dimensional viscous flow instabilities. We demonstrate that the approach can retain the high-order accuracy of the underlying numerical scheme even at smooth extrema, ensure the positivity of the species density and pressure in the vicinity of shocks and contact discontinuities, and accurately predict small-scale flow features with minimal numerical dissipation.

Positivity-preserving entropy filtering for the ideal magnetohydrodynamics equations

In this work, we present a positivity-preserving adaptive filtering approach for discontinuous spectral element approximations of the ideal magnetohydrodynamics equations. This approach combines the entropy filtering method (Dzanic and Witherden, 2022) for shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting approach is introduced to ensure that the positivity and entropy constraints remain satisfied by the discrete solution. Furthermore, a computationally efficient algorithm for solving the optimization process for this nonlinear filtering approach is presented. The resulting scheme can robustly resolve strong discontinuities on general unstructured grids without tunable parameters while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown in numerical experiments on various problems including extremely magnetized blast waves and three-dimensional magnetohydrodynamic instabilities.

An Explicit–Implicit Spectral Element Scheme for the Nonlinear Space Fractional Schrödinger Equation

In this paper, we solve the space fractional nonlinear Schrödinger equation (SFNSE) by developing an explicit–implicit spectral element scheme, which is formulated based on the Legendre spectral element approximation in space and the Crank–Nicolson leap frog (CNLF) difference discretization in time. Both mass and energy conservative properties are discussed for the spectral element scheme. Numerical stability and convergence of the scheme are proved. Numerical experiments are performed to confirm the high accuracy and efficiency of the proposed numerical scheme.

On the Anti-Aliasing Properties of Entropy Filtering for Discontinuous Spectral Element Approximations of Under-Resolved Turbulent Flows

For large Reynolds number flows, it is typically necessary to perform simulations that are under-resolved with respect to the underlying flow physics. For nodal discontinuous spectral element approximations of these under-resolved flows, the collocation projection of the nonlinear flux can introduce aliasing errors which can result in numerical instabilities. In Dzanic and Witherden (J. Comput. Phys., 468, 2022), an entropy-based adaptive filtering approach was introduced as a robust, parameter-free shock-capturing method for discontinuous spectral element methods. This work explores the ability of entropy filtering for mitigating aliasing-driven instabilities in the simulation of under-resolved turbulent flows through high-order implicit large eddy simulations of a NACA0021 airfoil in deep stall at a Reynolds number of 270,000. It was observed that entropy filtering can adequately mitigate aliasing-driven instabilities without degrading the accuracy of the underlying high-order scheme on par with standard anti-aliasing methods such as over-integration, albeit with marginally worse performance at higher approximation orders.

A priori and a posteriori error analysis of hp spectral element discretization for optimal control problems with elliptic equations

. A priori and a posteriori error estimates of hp spectral element method for an elliptic optimal control problem with L2-norm control constraint are investigated in this paper. We then present the optimality conditions and set up hp spectral element discretization scheme. Based on some important interpolation operators and suitable immediate variable, a priori error estimates for this problem in hp spectral element discretization are established carefully. Furthermore, using some interpolation operators, a rigorous posteriori error estimates of hp spectral element approximation are also proved for control and the coupled state approximation in L2−H1-norm and L2−L2-norm, respectively. Such estimators can be used to construct reliable adaptive spectral element methods for optimal control problems. Finally, the error analysis results are confirmed by numerical results.

A Posteriori Error Estimates for $hp$ Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints

A Posteriori Error Estimates for $hp$ Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints

High-order SAFE computation of reflection and transmission coefficients for functionally-graded poroelastic plates

This paper aims at studying the ultrasound response of a functionally-graded and anisotropic porous elastic layer surrounded by a fluid in the frequency domain. Based on Biot’s theory in a planar configuration, a semi-analytical high-order finite element method was proposed for computing the reflection and transmission coefficients of a plane wave with oblique incidence to the interface between the fluid and poroelastic layer. Two approximation techniques were investigated: (i) spectral element approximation; and (ii) isogeometric approximation. A numerical study was performed on the ultrasound response of bone materials with different depth-varied profiles of porosity showing high effectiveness of the proposed high-order methods.

A posteriori error estimates of hp spectral element method for parabolic optimal control problems

<abstract><p>In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $ L^2(H^1)-L^2(L^2) $ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $ L^2(L^2)-L^2(L^2) $ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.</p></abstract>

An error estimator for spectral method approximation of flow control with state constraint

<abstract><p>We consider the spectral approximation of flow optimal control problems with state constraint. The main contribution of this work is to derive an a posteriori error estimator, and show the upper and lower bounds for the approximation error. Numerical experiment shows the efficiency of the error indicator, which will be used to construct reliable adaptive <italic>hp</italic> spectral element approximation for flow control in the future work.</p></abstract>

SoftFEM: Revisiting the spectral finite element approximation of second-order elliptic operators

We propose, analyze mathematically, and study numerically a novel approach for the finite element approximation of the spectrum of second-order elliptic operators. The main idea is to reduce the stiffness of the problem by subtracting a least-squares penalty on the gradient jumps across the mesh interfaces from the standard stiffness bilinear form. This penalty bilinear form is similar to the known technique used to stabilize finite element approximations in various contexts. The penalty term is designed to dampen the high frequencies in the spectrum and so it is weighted here by a negative coefficient. The resulting approximation technique is called softFEM since it reduces the stiffness of the problem. The two key advantages of softFEM over the standard Galerkin FEM are to improve the approximation of the eigenvalues in the upper part of the discrete spectrum and to reduce the condition number of the stiffness matrix. We derive a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form. Then we prove that softFEM delivers the same optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. We next compare the discrete eigenvalues obtained when using Galerkin FEM and softFEM. Finally, a detailed analysis of linear softFEM for the 1D Laplace eigenvalue problem delivers a sensible choice for the softness parameter. With this choice, the stiffness reduction ratio scales linearly with the polynomial degree. Various numerical experiments illustrate the benefits of using softFEM over Galerkin FEM.

H(curl2)-conforming quadrilateral spectral element method for quad-curl problems

In this paper, we propose an [Formula: see text]-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. [Formula: see text]-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is worth noting that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our [Formula: see text]-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.

A Split-form, Stable CG/DG-SEM for Wave Propagation Modeled by Linear Hyperbolic Systems

We present a hybrid continuous and discontinuous Galerkin spectral element approximation that leverages the advantages of each approach. The continuous Galerkin approximation is used on interior element faces where the equation properties are continuous. A discontinuous Galerkin approximation is used at physical boundaries and if there is a jump in properties at a face. The approximation uses a split form of the equations and two-point fluxes to ensure stability for unstructured quadrilateral/hexahedral meshes with curved elements. The approximation is also conservative and constant state preserving on such meshes. Spectral accuracy is obtained for all examples, which include wave scattering at a discontinuous medium boundary.

Registration-Based Model Reduction in Complex Two-Dimensional Geometries

We present a general—i.e., independent of the underlying equation—egistration procedure for parameterized model order reduction. Given the spatial domain \(\varOmega \subset \mathbb {R}^2\) and the manifold \(\mathcal {M}= \{ u_{\mu } : \mu \in \mathcal {P} \}\) associated with the parameter domain \(\mathcal {P} \subset \mathbb {R}^P\) and the parametric field \(\mu \mapsto u_{\mu } \in L^2(\varOmega )\), our approach takes as input a set of snapshots \(\{ u^k \}_{k=1}^{n_\mathrm{train}} \subset \mathcal {M}\) and returns a parameter-dependent bijective mapping \({\varPhi }: \varOmega \times \mathcal {P} \rightarrow \mathbb {R}^2\): the mapping is designed to make the mapped manifold \(\{ u_{\mu } \circ {\varPhi }_{\mu }: \, \mu \in \mathcal {P} \}\) more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition \(\{ \varOmega _{q} \}_{q=1} ^{N_\mathrm{dd}}\) of the domain \(\varOmega \) such that \(\varOmega _1,\ldots ,\varOmega _{N_\mathrm{dd}}\) are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.

𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems

Abstract In this paper, we propose an H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral elements to solve the quad-curl problem on cubic meshes in three dimensions. Starting with generalized vectorial Jacobi polynomials, we first construct the basis functions of the H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral elements using the contravariant transform together with the affine mapping from the reference cube onto each physical element. Falling into four categories, interior modes, face modes, edge modes, and vertex modes, these H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming basis functions are constructed in an arbitrarily high degree with a hierarchical structure. Next, H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral element approximation schemes are established to solve the boundary value problem as well as the eigenvalue problem of quad-curl equations. Numerical experiments demonstrate the effectiveness and efficiency of the ℎ-version and the 𝑝-version of our H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral element method.

Spectral Element Methods a Priori and a Posteriori Error Estimates for Penalized Unilateral Obstacle Problem

The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre–Gauss–Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates.

Regularity and convergence analysis in Sobolev and H\xf6lder spaces for generalized Whittle\u2013Mat\xe9rn fields

We analyze several types of Galerkin approximations of a Gaussian random field mathscr {Z}:mathscr {D}times varOmega rightarrow mathbb {R} indexed by a Euclidean domain mathscr {D}subset mathbb {R}^d whose covariance structure is determined by a negative fractional power L^{-2beta } of a second-order elliptic differential operator L:= -nabla cdot (Anabla ) + kappa ^2. Under minimal assumptions on the domain mathscr {D}, the coefficients A:mathscr {D}rightarrow mathbb {R}^{dtimes d}, kappa :mathscr {D}rightarrow mathbb {R}, and the fractional exponent beta >0, we prove convergence in L_q(varOmega ; H^sigma (mathscr {D})) and in L_q(varOmega ; C^delta (overline{mathscr {D}})) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on H^{1+alpha }(mathscr {D})-regularity of the differential operator L, where 0<alpha le 1. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in L_{infty }(mathscr {D}times mathscr {D}) and in the mixed Sobolev space H^{sigma ,sigma }(mathscr {D}times mathscr {D}), showing convergence which is more than twice as fast compared to the corresponding L_q(varOmega ; H^sigma (mathscr {D}))-rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where Aequiv mathrm {Id}_{mathbb {R}^d} and kappa equiv {text {const.}}, and (b) an example of anisotropic, non-stationary Gaussian random fields in d=2 dimensions, where A:mathscr {D}rightarrow mathbb {R}^{2times 2} and kappa :mathscr {D}rightarrow mathbb {R} are spatially varying.

Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator. Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected noises are then numerically approximated with the finite element method. A general error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained. It is shown that with the proposed approach, convergence order of white noise driven SPDEs is improved by half for one-dimensional problems, and by an infinitesimal factor for higher-dimensional problems.

Hp spectral element approximation for integral state constrained optimal control problems governed by harmonic equations

In this paper, an integral state constrained optimal control problem governed by first bi-harmonic equation and its hp spectral element approximation are considered. We first present the optimality conditions of the continuous and discrete optimal control problems, respectively. Then, a priori error estimates of hp spectral element approximation for the optimal control problem are derived rigorously. Moreover, a posteriori error estimates are established for the coupled state and control approximations in detail. Such estimators can be used to construct reliable adaptive methods for optimal control problems.

A segregated spectral finite element method for the 2D transient incompressible Navier–Stokes equations

In this paper, the spectral element approximation and the velocity–pressuredecoupling method implementing the SIMPLE (semi-implicit method for pressure linked equations) algorithm are first combined to form a high-order segregated scheme for the solution of the two-dimensional transient incompressible Navier–Stokes equations. In contrast to previous segregated finite element methods based on the SIMPLE algorithm, the pressure equation is derived from the continuity equation using the element matrices to ensure convergence. High-order element basis functions are adopted, which greatly reduces the number of nodal points used in the calculation. The validation test that has an analytical solution demonstrates the high accuracy and convergence rate of the method. The flow in a lid-driven cavity with different inclination angles and the flow over a backward-facing step are investigated to further illustrate the performance of the scheme. The computed results are in excellent agreement with the benchmark solutions. The almost periodic solution for the flow of Re=10000 in a lid-driven square cavity is also captured by the present scheme.

$$C^1$$-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

In this talk, we propose a C¹-conforming quadrilateral spectral element method for fourth order partial differential equations. The approximation space is constituted by those globally C¹-continuous functions, which are essentially rational polynomials on each physical quadrilateral mapped from polynomials on the reference square by using bilinear transformations. So we first concentrate on the construction the shape functions of an arbitrary order on arbitrary quadrilateral meshes. The quadrilateral spectral element approximation scheme is then established with a brief explanation of the implementation. Numerical experiments are illustrated to demonstrate the accuracy and efficiency of our proposed method.