GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
A fast algorithm for particle simulations
Journal of Computational Physics
Laplace's equation and the Dirichlet-Neumann map in multiply connected domains
Journal of Computational Physics
Locally corrected multidimensional quadrature rules for singular functions
SIAM Journal on Scientific Computing
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Integral equation methods for Stokes flow and isotropic elasticity in the plane
Journal of Computational Physics
An Integral Equation Approach to the Incompressible Navier--Stokes Equations in Two Dimensions
SIAM Journal on Scientific Computing
Hybrid Gauss-Trapezoidal Quadrature Rules
SIAM Journal on Scientific Computing
A New Fast-Multipole Accelerated Poisson Solver in Two Dimensions
SIAM Journal on Scientific Computing
A new version of the fast multipole method for screened Coulomb interactions in three dimensions
Journal of Computational Physics
A fast solver for the Stokes equations with distributed forces in complex geometries
Journal of Computational Physics
An adaptive fast solver for the modified Helmholtz equation in two dimensions
Journal of Computational Physics
Fourth order accurate evaluation of integrals in potential theory on exterior 3D regions
Journal of Computational Physics
A fast method for solving the heat equation by layer potentials
Journal of Computational Physics
Krylov deferred correction accelerated method of lines transpose for parabolic problems
Journal of Computational Physics
A High-Order Solver for the Heat Equation in 1D domains with Moving Boundaries
SIAM Journal on Scientific Computing
High Order Accurate Methods for the Evaluation of Layer Heat Potentials
SIAM Journal on Scientific Computing
Fast integral equation methods for the modified Helmholtz equation
Journal of Computational Physics
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We present an efficient integral equation approach to solve the forced heat equation, u"t(x)-@Du(x)=F(x,u,t), in a two-dimensional, multiply-connected domain, with Dirichlet boundary conditions. Instead of using an integral equation formulation based on the heat kernel, we discretize in time, first. This approach, known as Rothe's method, leads to a non-homogeneous modified Helmholtz equation that is solved at each time step. We formulate the solution to this equation as a volume potential plus a double layer potential, and both of these potentials are calculated with available tools accelerated by the fast multipole method. For a total of N points in the discretization of the boundary and the domain, the total computational cost per time step is O(N). We demonstrate our approach on the heat equation and the Allen-Cahn equation.