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1. A convexity enforcing $${C}^{{0}}$$ interior penalty method for the Monge–Ampère equation on convex polygonal domains

   https://doi.org/10.1007/s00211-021-01210-x  

   Brenner, Susanne C. ; Sung, Li-yeng ; Tan, Zhiyu ; Zhang, Hongchao ( June 2021 , Numerische Mathematik)

   null (Ed.)

   Abstract We design and analyze a $$C^0$$ C 0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions. 

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2. The Carleman Contraction Mapping Method for Quasilinear Elliptic Equations with Over-determined Boundary Data

   https://doi.org/10.1007/s40306-023-00500-w  

   Nguyen, Loc H. ( April 2023 , Acta Mathematica Vietnamica)

   We propose a globally convergent numerical method to compute solutions to a general class of quasi-linear PDEs with both Neumann and Dirichlet boundary conditions. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the PDE under consideration. To find this fixed point, we define a recursive sequence with an arbitrary initial term using the same manner as in the proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution. On the other hand, we also show that our method delivers reliable solutions even when the given data are noisy. Numerical examples are presented. 

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3. A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

   https://doi.org/10.1007/s42967-021-00179-6  

   Li, Liang ; Zhu, Jun ; Shu, Chi-Wang ; Zhang, Yong-Tao ( March 2022 , Communications on Applied Mathematics and Computation)

   Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show high-order accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

    

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4. An efficient positive‐definite block‐preconditioned finite volume solver for two‐sided fractional diffusion equations on composite mesh

   https://doi.org/10.1002/nla.2372  

   Dai, Pingfei ; Jia, Jinhong ; Wang, Hong ; Wu, Qingbiao ; Zheng, Xiangcheng ( March 2021 , Numerical Linear Algebra with Applications)

   It is known that the solutions to space‐fractional diffusion equations exhibit singularities near the boundary. Therefore, numerical methods discretized on the composite mesh, in which the mesh size is refined near the boundary, provide more precise approximations to the solutions. However, the coefficient matrices of the corresponding linear systems usually lose the diagonal dominance and are ill‐conditioned, which in turn affect the convergence behavior of the iteration methods.In this work we study a finite volume method for two‐sided fractional diffusion equations, in which a locally refined composite mesh is applied to capture the boundary singularities of the solutions. The diagonal blocks of the resulting three‐by‐three block linear system are proved to be positive‐definite, based on which we propose an efficient block Gauss–Seidel method by decomposing the whole system into three subsystems with those diagonal blocks as the coefficient matrices. To further accelerate the convergence speed of the iteration, we use T. Chan's circulant preconditioner³¹as the corresponding preconditioners and analyze the preconditioned matrices' spectra. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method and its strong potential in dealing with ill‐conditioned problems. While we have not proved the convergence of the method in theory, the numerical experiments show that the proposed method is convergent.

    

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5. Traveltime Calculations for qP, qSV, and qSH Waves in Two‐Dimensional Tilted Transversely Isotropic Media

   https://doi.org/10.1029/2019JB018868  

   Huang, Guangnan ; Luo, Songting ; Deng, Juzhi ; Vavryčuk, Václav ( July 2020 , Journal of Geophysical Research: Solid Earth)

   This paper presents a fast sweeping method (FSM) to calculate the first‐arrival traveltimes of the qP, qSV, and qSH waves in two‐dimensional (2D) transversely isotropic media, whose symmetry axis may have an arbitrary orientation (tilted transverse isotropy [TTI]). The method discretizes the anisotropic eikonal equation with finite difference approximations on a rectangular mesh and solves the discretized system iteratively with the Gauss‐Seidel iterations along alternating sweeping orderings. At each mesh point, a highly nonlinear equation is solved to update the numerical solution until its convergence. For solving the nonlinear equation, an interval that contains the solutions is first determined and partitioned into few subintervals such that each subinterval contains one solution; then, the false position method is applied on these subintervals to compute the solutions; after that, among all possible solutions for the discretized equation, a causality condition is imposed, and the minimum solution satisfying the causality condition is chosen to update the solution. For problems with a point‐source condition, the FSM is extended for solving the anisotropic eikonal equation after a factorization technique is applied to resolve the source singularities, which yields clean first‐order accuracy. When dealing with the triplication of the qSV wave, solutions corresponding to the minimal group velocity are chosen such that continuous solutions are computed. The accuracy, efficiency, and capability of the proposed method are demonstrated with numerical experiments.

    

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