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1. A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

   https://doi.org/10.1090/cams/39  

   Brenner, Susanne; Sung, Li-yeng; Tan, Zhiyu; Zhang, Hongchao (October 2024, Communications of the American Mathematical Society)

   We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in ${\mathbb{R}}^2$. It is based on an isoparametric $C^0$finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions.A priorianda posteriorierror estimates together with corroborating numerical results are presented. 

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2. convergence of a homotopy finite element method for computing steady states of Burgers' equation

   https://doi.org/10.1051/m2an/2018046  

   Yang, Yong; Hao, Wenrui (January 2018, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper, the convergence of a homotopy method (1.1) for solving the steady state problem of Burgers’ equation is considered. When ν is fixed, we prove that the solution of (1.1) converges to the unique steady state solution as epsilon → 0, which is independent of the initial conditions. Numerical examples are presented to confirm this conclusion by using the continuous finite element method. In contrast, when ν = epsilon → 0, numerically we show that steady state solutions obtained by (1.1) indeed depend on initial conditions. 

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3. A C ^α finite difference method for the Caputo time‐fractional diffusion equation

   https://doi.org/10.1002/num.22686  

   Davis, Wesley; Noren, Richard; Shi, Ke (November 2020, Numerical Methods for Partial Differential Equations)

   Abstract We begin with a treatment of the Caputo time‐fractional diffusion equation, by using the Laplace transform, to obtain a Volterra integro‐differential equation. We derive and utilize a numerical scheme that is derived in parallel to the L1‐method for the time variable and a standard fourth‐order approximation in the spatial variable. The main method derived in this article has a rate of convergence ofO(k^α + h⁴)foru(x,t) ∈ C^α([0,T];C⁶(Ω)),0 < α < 1, which improves previous regularity assumptions that requireC²[0,T]regularity in the time variable. We also present a novel alternative method for a first‐order approximation in time, under a regularity assumption ofu(x,t) ∈ C¹([0,T];C⁶(Ω)), while exhibiting order of convergence slightly more thanO(k)in time. This allows for a much wider class of functions to be analyzed which was previously not possible under the L1‐method. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques. 

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4. OrbitN: A Symplectic Integrator for Planetary Systems Dominated by a Central Mass—Insight into Long-term Solar System Chaos

   https://doi.org/10.3847/1538-3881/acd63b  

   Zeebe, Richard E. (June 2023, The Astronomical Journal)

   Abstract Reliable studies of the long-term dynamics of planetary systems require numerical integrators that are accurate and fast. The challenge is often formidable because the chaotic nature of many systems requires relative numerical error bounds at or close to machine precision (∼10⁻¹⁶, double-precision arithmetic); otherwise, numerical chaos may dominate over physical chaos. Currently, the speed/accuracy demands are usually only met by symplectic integrators. For example, the most up-to-date long-term astronomical solutions for the solar system in the past (widely used in, e.g., astrochronology and high-precision geological dating) have been obtained using symplectic integrators. However, the source codes of these integrators are unavailable. Here I present the symplectic integratororbitN(lean version 1.0) with the primary goal of generating accurate and reproducible long-term orbital solutions for near-Keplerian planetary systems (here the solar system) with a dominant massM₀. Among other features,orbitN-1.0includesM₀’s quadrupole moment, a lunar contribution, and post-Newtonian corrections (1PN) due toM₀(fast symplectic implementation). To reduce numerical round-off errors, Kahan compensated summation was implemented. I useorbitNto provide insight into the effect of various processes on the long-term chaos in the solar system. Notably, 1PN corrections have the opposite effect on chaoticity/stability on a 100 Myr versus Gyr timescale. For the current application,orbitNis about as fast as or faster (factor 1.15–2.6) than comparable integrators, depending on hardware.¹¹The orbitN source code (C) is available athttp://github.com/rezeebe/orbitN. 

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5. A C 0 finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain

   https://doi.org/10.1093/imanum/drac026  

   Li, Hengguang; Yin, Peimeng; Zhang, Zhimin (July 2022, IMA Journal of Numerical Analysis)

   Abstract In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings. 

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