MADNESS

Solving PDEs in irregular geometries with multiresolution methods. I: Embedded Dirichlet boundary conditions In this work, we develop and analyze a formalism for solving boundary value problems in arbitrarily-shaped domains using the MADNESS (multiresolution adaptive numerical environment for scientific simulation) package for adaptive computation with multiresolution algorithms. We begin by implementing a previously-reported diffuse domain approximation for embedding the domain of interest into a larger domain (Li et al., 2009 [1]). Numerical and analytical tests both demonstrate that this approximation yields non-physical solutions with zero first and second derivatives at the boundary. This excessive smoothness leads to large numerical cancellation and confounds the dynamically-adaptive, multiresolution algorithms inside { t MADNESS}. We thus generalize the diffuse domain approximation, producing a formalism that demonstrates first-order convergence in both near- and far-field errors. We finally apply our formalism to an electrostatics problem from nanoscience with characteristic length scales ranging from 0.0001 to 300 nm.


References in zbMATH (referenced in 7 articles , 1 standard article )

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  1. Springer, Paul; Bientinesi, Paolo: Design of a high-performance GEMM-like tensor-tensor multiplication (2018)
  2. Burger, Martin; Elvetun, Ole Løseth; Schlottbom, Matthias: Analysis of the diffuse domain method for second order elliptic boundary value problems (2017)
  3. Paul Springer, Tong Su, Paolo Bientinesi: HPTT: A High-Performance Tensor Transposition C++ Library (2017) arXiv
  4. Harrison, Robert J.; Beylkin, Gregory; Bischoff, Florian A.; Calvin, Justus A.; Fann, George I.; Fosso-Tande, Jacob; Galindo, Diego; Hammond, Jeff R.; Hartman-Baker, Rebecca; Hill, Judith C.; Jia, Jun; Kottmann, Jakob S.; Yvonne Ou, M.-J.; Pei, Junchen; Ratcliff, Laura E.; Reuter, Matthew G.; Richie-Halford, Adam C.; Romero, Nichols A.; Sekino, Hideo; Shelton, William A.; Sundahl, Bryan E.; Thornton, W. Scott; Valeev, Edward F.; Vázquez-Mayagoitia, Álvaro; Vence, Nicholas; Yanai, Takeshi; Yokoi, Yukina: MADNESS: a multiresolution, adaptive numerical environment for scientific simulation (2016)
  5. Schlottbom, Matthias: Error analysis of a diffuse interface method for elliptic problems with Dirichlet boundary conditions (2016)
  6. Lervåg, Karl Yngve; Lowengrub, John: Analysis of the diffuse-domain method for solving PDEs in complex geometries (2015)
  7. Reuter, Matthew G.; Hill, Judith C.; Harrison, Robert J.: Solving PDEs in irregular geometries with multiresolution methods. I: Embedded Dirichlet boundary conditions (2012)