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1. Fast Poisson solvers for spectral methods

   https://doi.org/10.1093/imanum/drz034  

   Fortunato, Daniel; Townsend, Alex (November 2019, IMA Journal of Numerical Analysis)

   Abstract Poisson’s equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference (FD) and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here we derive spectral methods for solving Poisson’s equation on a square, cylinder, solid sphere and cube that have optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom used to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of FD matrices, our solver exploits a separated spectra property that holds for our carefully designed spectral discretizations. Without parallelization we can solve Poisson’s equation on a square with 100 million degrees of freedom in under 2 min on a standard laptop. 

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2. 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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3. Multi-discretization domain specific language and code generation for differential equations

   https://doi.org/10.1016/j.jocs.2023.101981  

   Heisler, Eric; Deshmukh, Aadesh; Mazumder, Sandip; Sadayappan, Ponnuswamy; Sundar, Hari (April 2023, Journal of computational science)

   Finch, a domain specific language and code generation framework for partial differential equations (PDEs), is demonstrated here to solve two classical problems: steady-state advection diffusion equation (single PDE) and the phonon Boltzmann transport equation (coupled PDEs). Both finite volume and finite element methods are explored. In addition to work presented at the 2022 International Conference on Computational Science (Heisler et al., 2022), we include recent developments for solving nonlinear equations using both automatic and symbolic differentiation, and demonstrate the capability for the Bratu (nonlinear Poisson) equation. 

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4. Geometric transformation of finite element methods: Theory and applications

   https://doi.org/10.1016/j.apnum.2023.07.002  

   Holst, Michael; Licht, Martin (October 2023, Applied Numerical Mathematics)

   We present a new technique to apply finite element methods to partial differential equations over curved domains. A change of variables along a coordinate transformation satisfying only low regularity assumptions can translate a Poisson problem over a curved physical domain to a Poisson problem over a polyhedral parametric domain. This greatly simplifies both the geometric setting and the practical implementation, at the cost of having globally rough non-trivial coefficients and data in the parametric Poisson problem. Our main result is that a recently developed broken Bramble-Hilbert lemma is key in harnessing regularity in the physical problem to prove higher-order finite element convergence rates for the parametric problem. Numerical experiments are given which confirm the predictions of our theory. 

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5. A new div-div-conforming symmetric tensor finite element space with applications to the biharmonic equation

   https://doi.org/10.1090/mcom/3957  

   Chen, Long; Huang, Xuehai (March 2024, Mathematics of Computation)

   A new H(divdiv)-conforming finite element is presented, which avoids the need for supersmoothness by redistributing the degrees of freedom to edges and faces. This leads to a hybridizable mixed method with superconvergence for the biharmonic equation. Moreover, new finite element divdiv complexes are established. Finally, new weak Galerkin and C0 discontinuous Galerkin methods for the biharmonic equation are derived. 

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