Software and High-Performance Computing

Computing the Distance between Two Finite Element Solutions Defined on Different 3D Meshes on a GPU


This article introduces a new method to efficiently compute the distance (i.e., $L^p$ norm of the difference) between two functions supported by two different meshes of the same 3D domain. The functions that we consider are typically finite element solutions discretized in different function spaces supported by meshes that are potentially completely unrelated. Our method computes an approximation of the distance by resampling both fields over a set of parallel 2D regular grids. By leveraging the parallel horse power of computer graphics hardware (graphics processing unit (GPU)), our method can efficiently compute distances between meshes with multimillion elements in seconds. We demonstrate our method applied to different problems (distance between known functions, Poisson solutions, and linear elasticity solutions) using different function spaces (Lagrange polynomials from order one to seven) and different meshes (tetrahedral and hexahedral, with linear or quadratic geometry).


  1. distance
  2. field distance
  3. finite element
  4. error estimate
  5. mesh comparison
  6. approximation error
  7. error analysis

MSC codes

  1. 65D05
  2. 65D30
  3. 65N30
  4. 68U20

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Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Computing the distance between two finite element solutions defined on different 3D meshes on a GPU

Authors: Maxence Reberol, Bruno Levy


Type: Compressed code files

Contents: Source code of the implementation referenced in the paper and used to generate the results.

Justification: The source code of our implementation is useful to anyone who would like to use our program or to reproduce and extend our results.


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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: C131 - C155
ISSN (online): 1095-7197


Submitted: 13 February 2017
Accepted: 14 December 2017
Published online: 20 February 2018


  1. distance
  2. field distance
  3. finite element
  4. error estimate
  5. mesh comparison
  6. approximation error
  7. error analysis

MSC codes

  1. 65D05
  2. 65D30
  3. 65N30
  4. 68U20



Funding Information

H2020 European Research Council : StG-2012-307877

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