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| BiomathematicsMathematical modeling of immune-cardiovascular-endocrine dynamics following endotoxin administrationKristen Windoloski Department of Mathematics, North Carolina State University |

| Algebra and Number TheoryNo Seminar |

| Applied Mathematics and Machine LearningRecent Progress on Q^k Spectral Element Method: Accuracy, Monotonicity and ApplicationsXiangxiong Zhang Department of Mathematics, Purdue University |

1) Accuracy: when the least accurate (k+1)-point Gauss-Lobatto quadrature is used, the spectral element method is also a finite difference (FD) scheme, and this FD scheme can sometimes be (k+2)-th order accurate for k>=2. This has been observed in practice but never proven before as rigorous a priori error estimates. We are able to prove it for linear elliptic, wave, parabolic and SchrĂ¶dinger equations for Dirichlet boundary conditions. For Neumann boundary conditions, (k+2)-th order can be proven if there is no mixed second order derivative. Otherwise, only (k+3/2)-th order can be proven and some order loss is indeed observed in numerical tests. The accuracy result also applies to spectral element method on any curvilinear mesh that can be smoothly mapped to a rectangular mesh, e.g., solving a wave equation on an annulus region with a curvilinear mesh generated by polar coordinates.

2) Monotonicity: consider solving the Poisson equation, then a scheme is called monotone if the inverse of the stiffness matrix is entrywise non-negative. It is well known that second order centered difference or P1 finite element method can form an M-matrix thus they are monotone, and high order accurate schemes in general are not monotone. But on structured meshes, high order accurate schemes can be monotone, though they do not form M-matrices. In particular, we have proven that the fourth order accurate FD scheme (Q^2 spectral element method) is a product of two M-matrices thus monotone for a variable coefficient diffusion operator: this is the first time that a high order accurate scheme is proven monotone for a variable coefficient operator. We have also proven the fifth order accurate FD scheme (Q^3 spectral element method) is a product of three M-matrices thus monotone for the Poisson equation: this is the first time that a fifth order accurate discrete Laplacian is proven monotone in two dimensions (all previously known high order monotone discrete Laplacian in 2D are fourth order accurate).

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Meeting ID: 976 3095 1027

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