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1. Efficient quadrature rules for finite element discretizations of nonlocal equations

   Eugenio Aulisa, Giacomo Capodaglio (November 2022, Numerical methods for partial differential equations)

   In this paper, we design efficient quadrature rules for finite element (FE) discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three-dimensional settings. In this work, we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the “mollified” solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two- and three-dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient FE assembly. 

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2. A high-order shock capturing discontinuous Galerkin–finite difference hybrid method for GRMHD

   https://doi.org/10.1088/1361-6382/ac8864  

   Deppe, Nils; Hébert, François; Kidder, Lawrence E; Teukolsky, Saul A (August 2022, Classical and Quantum Gravity)

   Abstract We present a discontinuous Galerkin (DG)–finite difference (FD) hybrid scheme that allows high-order shock capturing with the DG method for general relativistic magnetohydrodynamics. The hybrid method is conceptually quite simple. An unlimited DG candidate solution is computed for the next time step. If the candidate solution is inadmissible, the time step is retaken using robust FD methods. Because of its a posteriori nature, the hybrid scheme inherits the best properties of both methods. It is high-order with exponential convergence in smooth regions, while robustly handling discontinuities. We give a detailed description of how we transfer the solution between the DG and FD solvers, and the troubled-cell indicators necessary to robustly handle slow-moving discontinuities and simulate magnetized neutron stars. We demonstrate the efficacy of the proposed method using a suite of standard and very challenging 1D, 2D, and 3D relativistic magnetohydrodynamics test problems. The hybrid scheme is designed from the ground up to efficiently simulate astrophysical problems such as the inspiral, coalescence, and merger of two neutron stars. 

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3. Compactness results for a Dirichlet energy of nonlocal gradient with applications

   https://doi.org/10.1002/num.23149  

   Han, Zhaolong; Mengesha, Tadele; Tian, Xiaochuan (November 2024, Numerical Methods for Partial Differential Equations)

   Abstract We prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem. 

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4. Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

   https://doi.org/10.1093/imanum/drae094  

   Imbert-Gérard, Lise-Marie; Moiola, Andrea; Perinati, Chiara; Stocker, Paul (January 2025, IMA Journal of Numerical Analysis)

   Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems. 

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5. OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

   https://doi.org/10.1090/mcom/3998  

   Peng, Manting; Sun, Zheng; Wu, Kailiang (July 2024, Mathematics of Computation)

   Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess bothscale-invariantandevolution-invariantproperties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate fornonlinearDG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure isnonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control.It reveals the role of the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques.Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method. 

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