An Adaptive High Order Direct Solution Technique for Elliptic Boundary Value Problems

Geldermans, Peter; Gillman, Adrianna

doi:10.1137/17m1156320

Cited by 10 publications

(17 citation statements)

References 29 publications

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“…collocation technique, first presented in [6], is ideal for the HPS method because it does not involve corner discretization points, for which Poincaré-Steklov operators are not always well defined. The modified spectral collocation technique begins with the classic n c × n c product Chebychev grid and the corresponding standard spectral differential matrices D x and D y , as defined in [4].…”

Section: Leaf Computationmentioning

confidence: 99%

A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems

Beams

Gillman

Hewett

2020

Computers & Mathematics with Applications

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The recently developed Hierarchical Poincaré-Steklov (HPS) method is a highorder discretization technique that comes with a direct solver. Results from previous papers demonstrate the method's ability to solve Helmholtz problems to high accuracy without the so-called pollution effect. While the asymptotic scaling of the direct solver's computational cost is the same as the nested dissection method, serial implementations of the solution technique are not practical for large scale numerical simulations. This manuscript presents the first parallel implementation of the HPS method. Specifically, we introduce an approach for a shared memory implementation of the solution technique utilizing parallel linear algebra. This approach is the foundation for future large scale simulations on supercomputers and clusters with large memory nodes. Performance results on a desktop computer (resembling a large memory node) are presented.

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Section: Leaf Computationmentioning

confidence: 99%

A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems

Beams

Gillman

Hewett

2020

Computers & Mathematics with Applications

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“…The essential difficulty that arises is that when boxes of different sizes are joined, the collocation nodes along the joint boundary will not align. It is demonstrated in [1,5] that this difficulty can stably and efficiently be handled by incorporating local interpolation operators.…”

Section: Discretizationmentioning

confidence: 99%

“…In the HPS algorithm the PDE is enforced on interior nodes and continuity of the normal derivative is enforced on the leaf boundary. Now, due to the structure of the update formula (5), if at some time u n has an error component in the null space of the operator that is used to solve for a slope k i , then this will remain throughout the solution process. Although this does not affect the stability of the method it may result in loss of relative accuracy as the solution evolves.…”

Section: Neumann Data Correction In the Slope Formulationmentioning

confidence: 99%

HPS Accelerated Spectral Solvers for Time Dependent Problems: Part I, Algorithms

Babb

Martinsson

Appelö

2020

Lecture Notes in Computational Science and Engineering

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A high-order convergent numerical method for solving linear and non-linear parabolic PDEs is presented. The time-stepping is done via an explicit, singly diagonally implicit Runge–Kutta (ESDIRK) method of order 4 or 5, and for the implicit solve, we use the recently developed “Hierarchial Poincaré–Steklov (HPS)” method. The HPS method combines a multidomain spectral collocation discretization technique (a “patching method”) with a nested-dissection type direct solver. In the context under consideration, the elliptic solve required in each time-step involves the same coefficient matrix, which makes the use of a direct solver particularly effective. In this chapter is the first in a two part sequence; the first part describes the methodology, the second part presents numerical experiments.

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“…Recent progress in the development of fast and flexible spectral element methods for second-order partial differential equations [16,17,18,19,20] provides an interesting opportunity to extend the capabilities of low-temperature plasma simulations. Indeed, the method based on the hierarchical Poincaré -Steklov (HPS) scheme, first proposed by Martinsson in [16], has a number of attractive features.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, the method based on the hierarchical Poincaré -Steklov (HPS) scheme, first proposed by Martinsson in [16], has a number of attractive features. Its complexity is comparable to that of the existing direct solvers for sparse linear systems [16,17], it is able to treat complex geometries [16,20], enables the use of adaptive mesh refinement [19] and is easy to implement and parallelize. In view of its application to modelling of low temperature plasmas, this method can be used for computing the electric potential, photoionization source terms and solving the transport equations for plasma components.…”

Section: Introductionmentioning

confidence: 99%

A spectral element method for modelling streamer discharges in low-temperature atmospheric-pressure plasmas

Semenov,

Weltmann

2021

Preprint

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Streamers are ionization fronts that occur in gases at atmospheric and sub-atmospheric pressures. Numerical studies of streamers are important for practical applications but are challenging due to the multiscale nature of this discharge type. This paper introduces a spectral element method for modelling streamer discharges. The method is developed for Cartesian grids but can be extended to be used on unstructured meshes. The streamer model is based on the Poisson equation for the electric potential and the electron continuity equation. The Poisson equation is discretized via a spectral method based on the integral representation of the solution. The hierarchical Poincaré -Steklov (HPS) scheme is used to solve the resulting set of equations. The electron continuity equation is solved by means of the discontinuous Galerkin spectral element method (DGSEM). The DGSEM is extended by an alternative definition of the diffusion flux. A subcell finite volume method is used to stabilize the DGSEM scheme, if required. The entire simulation scheme is validated by solving a number of test problems and reproducing the results of previous studies. Adaptive mesh refinement is used to reduce the number of unknowns. The proposed method is found to be sufficiently fast for being used in practical applications. The flexibility of the method provides an interesting opportunity to broaden the range of problems that can be addressed in numerical studies of low-temperature plasma discharges.

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An Adaptive High Order Direct Solution Technique for Elliptic Boundary Value Problems

Cited by 10 publications

References 29 publications

A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems

A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems

HPS Accelerated Spectral Solvers for Time Dependent Problems: Part I, Algorithms

A spectral element method for modelling streamer discharges in low-temperature atmospheric-pressure plasmas

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