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1. Superconvergence of High Order Finite Difference Schemes Based on Variational Formulation for Elliptic Equations

   https://doi.org/https://doi.org/10.1007/s10915-020-01144-w  

   Li, Hao ; Zhang, Xiangxiong ( January 2020 , Journal of scientific computing)

   The classical continuous finite element method with Lagrangian Q^k basis reduces to a finite difference scheme when all the integrals are replaced by the (𝑘+1)×(𝑘+1) Gauss–Lobatto quadrature. We prove that this finite difference scheme is (𝑘+2)-th order accurate in the discrete 2-norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values. We also give a convenient implementation for the case 𝑘=2, which is a simple fourth order accurate elliptic solver on a rectangular domain. 

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2. Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using P k elements

   https://doi.org/10.1051/m2an/2019080  

   Liu, Yong ; Shu, Chi-Wang ; Zhang, Mengping ( March 2020 , ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using P k elements on uniform Cartesian meshes, and prove that the error in the L 2 norm achieves optimal ( k  + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k  ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k  = 0, 1, 2, 3 and k  = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples. 

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3. Fast Barycentric-Based Evaluation Over Spectral/hp Elements

   https://doi.org/10.1007/s10915-021-01750-2  

   Laughton, Edward ; Zala, Vidhi ; Narayan, Akil ; Kirby, Robert M. ; Moxey, David ( January 2022 , Journal of Scientific Computing)

   As the use of spectral/hpelement methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendous attention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/hpelement methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/hpelement libraryNektar++by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrix-multiplication applied to evaluate a point at a given location. We present results from a rigorous series of benchmarking tests for a variety of element shapes, polynomial orders and dimensions. We show that when the point of interest is to be repeatedly evaluated, the barycentric method performs at worst$$50\%$$$50\%$slower, when compared to a cached matrix evaluation. However, when the point of interest changes repeatedly so that the interpolation matrix must be regenerated in the ‘standard’ approach, the barycentric method yields far greater performance, with a minimum speedup factor of$$7\times $$$7\times$. Furthermore, when derivatives of the solution evaluation are also required, the barycentric method in general slightly outperforms the cached interpolation matrix method across all elements and orders, with an up to$$30\%$$$30\%$speedup. Finally we investigate a real-world example of scalar transport using a non-conformal discontinuous Galerkin simulation, in which we observe around$$6\times $$$6\times$speedup in computational time for the barycentric method compared to the matrix-based approach. We also explore the complexity of both interpolation methods and show that the barycentric interpolation method requires$${\mathcal {O}}(k)$$$O\left(k\right)$storage compared to a best case space complexity of$${\mathcal {O}}(k^2)$$$O\left(k^2\right)$for the Lagrangian interpolation matrix method.

    

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4. Embedding properties of network realizations of dissipative reduced order models

   Druskin, Guddati ( July 2022 , HOUSEHOLDER SYMPOSIUM XXI)

   Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and block-tridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistor-capacitor-inductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a non-symmetric Lanczos algorithm written in J-symmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nite-dierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich data-manifolds), a nite-dierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in port-Hamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible non-positivity of conductors for the non-Stieltjes case, we introduce an additional complex s-dependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on Krein-Nudelman theory [5], that results in special data-driven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the three-point second dierences: I. A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back 

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5. Two‐order superconvergence for a weak Galerkin method on rectangular and cuboid grids

   https://doi.org/10.1002/num.22918  

   Wang, Junping ; Wang, Xiaoshen ; Ye, Xiu ; Zhang, Shangyou ; Zhu, Peng ( September 2022 , Numerical Methods for Partial Differential Equations)

   This article introduces a particular weak Galerkin (WG) element on rectangular/cuboid partitions that uses th order polynomial for weak finite element functions and th order polynomials for weak derivatives. This WG element is highly accurate with convergence two orders higher than the optimal order in an energy norm and the norm. The superconvergence is verified analytically and numerically. Furthermore, the usual stabilizer in the standard weak Galerkin formulation is no longer needed for this element.

    

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