### Abstract

We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a black-box fashion; (2) it is second-order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the embedded boundary integral method (EBI), is based on Anita Mayo's work for the Poisson's equation: "The Fast Solution of Poisson's and the Biharmonic Equations on Irregular Regions", SIAM Journal on Numerical Analysis, 21 (1984) 285-299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström's method. The rectangular domain problem is discretized by finite elements for a velocity-pressure formulation with equal order interpolation bilinear elements (Q1-Q1). Stabilization is used to circumvent the inf-sup condition for the pressure space. For the integral equations, fast matrix-vector multiplications are achieved via an N log N algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsity low-rank blocks. The regular grid solver is a Krylov method (conjugate residuals) combined with an optimal two-level Schwartz-preconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates.

Original language | English (US) |
---|---|

Pages (from-to) | 317-348 |

Number of pages | 32 |

Journal | Journal of Computational Physics |

Volume | 193 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2004 |

### Fingerprint

### Keywords

- Cartesian grid methods
- Double-layer potential
- Embedded domain methods
- Fast multipole methods
- Fast solvers
- Fictitous domain methods
- Immersed interface methods
- Integral equations
- Moving boundaries
- Stokes equations

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)

### Cite this

*Journal of Computational Physics*,

*193*(1), 317-348. https://doi.org/10.1016/j.jcp.2003.08.011

**A fast solver for the Stokes equations with distributed forces in complex geometries.** / Biros, George; Ying, Lexing; Zorin, Denis.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 193, no. 1, pp. 317-348. https://doi.org/10.1016/j.jcp.2003.08.011

}

TY - JOUR

T1 - A fast solver for the Stokes equations with distributed forces in complex geometries

AU - Biros, George

AU - Ying, Lexing

AU - Zorin, Denis

PY - 2004/1/1

Y1 - 2004/1/1

N2 - We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a black-box fashion; (2) it is second-order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the embedded boundary integral method (EBI), is based on Anita Mayo's work for the Poisson's equation: "The Fast Solution of Poisson's and the Biharmonic Equations on Irregular Regions", SIAM Journal on Numerical Analysis, 21 (1984) 285-299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström's method. The rectangular domain problem is discretized by finite elements for a velocity-pressure formulation with equal order interpolation bilinear elements (Q1-Q1). Stabilization is used to circumvent the inf-sup condition for the pressure space. For the integral equations, fast matrix-vector multiplications are achieved via an N log N algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsity low-rank blocks. The regular grid solver is a Krylov method (conjugate residuals) combined with an optimal two-level Schwartz-preconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates.

AB - We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a black-box fashion; (2) it is second-order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the embedded boundary integral method (EBI), is based on Anita Mayo's work for the Poisson's equation: "The Fast Solution of Poisson's and the Biharmonic Equations on Irregular Regions", SIAM Journal on Numerical Analysis, 21 (1984) 285-299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström's method. The rectangular domain problem is discretized by finite elements for a velocity-pressure formulation with equal order interpolation bilinear elements (Q1-Q1). Stabilization is used to circumvent the inf-sup condition for the pressure space. For the integral equations, fast matrix-vector multiplications are achieved via an N log N algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsity low-rank blocks. The regular grid solver is a Krylov method (conjugate residuals) combined with an optimal two-level Schwartz-preconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates.

KW - Cartesian grid methods

KW - Double-layer potential

KW - Embedded domain methods

KW - Fast multipole methods

KW - Fast solvers

KW - Fictitous domain methods

KW - Immersed interface methods

KW - Integral equations

KW - Moving boundaries

KW - Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=0344981320&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0344981320&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2003.08.011

DO - 10.1016/j.jcp.2003.08.011

M3 - Article

AN - SCOPUS:0344981320

VL - 193

SP - 317

EP - 348

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -