Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Conley, Rebecca; Delaney, Tristan J.; Jiao, Xiangmin

doi:10.1002/nme.5246

Cited by 5 publications

(2 citation statements)

References 48 publications

(99 reference statements)

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“…For the reconstruction over tetrahedral meshes in 3-D, the standard k-ring neighbors grow very rapidly. To allow finer granularity, we define k-ring neighbor cells with 1/3-ring increments, similar to those defined in [32]:…”

Section: Generalization Of Wls-eno Schemes To 2-d and 3-dmentioning

confidence: 99%

WLS-ENO: Weighted-least-squares based essentially non-oscillatory schemes for finite volume methods on unstructured meshes

Liu

Jiao

2016

Journal of Computational Physics

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ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes are widely used high-order schemes for solving partial differential equations (PDEs), especially hyperbolic conservation laws with piecewise smooth solutions. For structured meshes, these techniques can achieve high order accuracy for smooth functions while being non-oscillatory near discontinuities. For unstructured meshes, which are needed for complex geometries, similar schemes are required but they are much more challenging. We propose a new family of non-oscillatory schemes, called WLS-ENO, in the context of solving hyperbolic conservation laws using finite-volume methods over unstructured meshes. WLS-ENO is derived based on Taylor series expansion and solved using a weighted least squares formulation. Unlike other non-oscillatory schemes, the WLS-ENO does not require constructing sub-stencils, and hence it provides more flexible framework and is less sensitive to mesh quality. We present rigorous analysis of the accuracy and stability of WLS-ENO, and present numerical results in 1-D, 2-D, and 3-D for a number of benchmark problems, and also report some comparisons against WENO.

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Section: Generalization Of Wls-eno Schemes To 2-d and 3-dmentioning

confidence: 99%

WLS-ENO: Weighted-least-squares based essentially non-oscillatory schemes for finite volume methods on unstructured meshes

Liu

Jiao

2016

Journal of Computational Physics

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“…The matrices from the GFDM are always nonsymmetric. A close method is the AES‐FEM, of which the test functions are similar to those of the FEM but the basis functions are similar to those of the GFDM. The linear systems from the AES‐FEM are also nonsymmetric.…”

Section: Introductionmentioning

confidence: 99%

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Ghai

Jiao

2018

Numerical Linear Algebra App

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Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss-Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts.BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill-conditioned or saddle-point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.

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Preserving superconvergence of spectral elements for curved domains via h- and p-geometric refinement

Jones,

Conley,

Jiao

2023

Engineering with Computers

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Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Cited by 5 publications

References 48 publications

WLS-ENO: Weighted-least-squares based essentially non-oscillatory schemes for finite volume methods on unstructured meshes

WLS-ENO: Weighted-least-squares based essentially non-oscillatory schemes for finite volume methods on unstructured meshes

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Preserving superconvergence of spectral elements for curved domains via h- and p-geometric refinement

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