An unfitted boundary algebraic equation method with Calderón preconditioning for 2D Stokes flow in irregular geometry

An unfitted boundary algebraic equation method based on difference potentials and lattice Green’s function in 3D

An unfitted boundary algebraic equation method with static-dynamic reduction for evolving implicit geometries

Repeated elliptic solves on domains with evolving boundaries arise in moving-interface simulation, design, and reactive navigation. Even when a fixed Cartesian grid avoids remeshing, rebuilding all boundary interactions for every configuration can limit the efficiency of repeated solves. We develop a static--dynamic boundary reduction for an unfitted lattice Green's function method on prescribed moving planar domains. Like boundary integral and boundary element methods, the formulation reduces the problem to boundary-supported unknowns through a Green representation. Its construction, however, reverses the usual order: the Cartesian operator is discretized before the Green representation is formed, rather than representing the continuous problem first and then discretizing the boundary. This discretize-then-represent viewpoint avoids boundary meshes and singular quadrature. The method also separates interactions associated with stationary geometry from those affected by motion, reuses the invariant part throughout a simulation, and updates only couplings involving the changing boundary. Boundary conditions are imposed at true interface intersections, lattice-kernel data are reused, and the interior field is reconstructed by a fast sine-transform solver. The principal contribution is an implemented and validated update strategy for translating, deforming, appearing, and topology-changing obstacles.

A Fully Adaptive, High-Order, Fast Poisson Solver for Multiscale Smooth Two-Dimensional Geometries

Abstract. We present a new framework for the fast solution of inhomogeneous elliptic boundary value problems in domains with smooth boundaries. High-order solvers based on adaptive box codes or the fast Fourier transform can efficiently treat the volumetric inhomogeneity but require care to be taken near the boundary to ensure that the volume data is globally smooth. We avoid function extension or cut-cell quadratures near the boundary by dividing the domain into two regions: a bulk region away from the boundary that is efficiently treated with a truncated free-space box code, and a variable-width boundary-conforming strip region that is treated with a spectral collocation method and an accompanying fast direct solver. Particular solutions in each region are then combined with Laplace layer potentials to yield the global solution. The resulting solver has an optimal computational complexity of [Formula: see text] for an adaptive discretization with [Formula: see text] degrees of freedom. With an efficient two-dimensional (2D) implementation we demonstrate adaptive resolution of volumetric data, boundary data, and geometric features across a wide range of length scales, to typically 10-digit accuracy. The cost of all boundary corrections remains small relative to that of the bulk box code. The extension to 3D is expected to be straightforward in many cases because the strip “thickens” an existing boundary quadrature. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/danfortunato/fully-adaptive-poisson and in the supplementary materials ( fully-adaptive-poisson-master.zip [170KB]). [Formula: see text]

Solving the Incompressible Navier-Stokes Equations on Oriented Curved Surfaces Discretized by Point Clouds

We present a meshfree numerical solver for the incompressible Navier-Stokes equations on oriented curved surfaces that are represented by surface point clouds. On curved surfaces, numerical challenges pertaining to stiffness and pressure-velocity coupling are exacerbated. Moreover, vector calculus on curved surfaces differs from its Euclidean counterpart. The presented method operates on surface point clouds in an Eulerian frame of reference without requiring a computational grid or mesh. It achieves consistent approximation in space and time with high order of accuracy; we demonstrate up to order six. The incompressibility constraint is locally imposed as a weak artificial compressibility approximation, avoiding global matrix inversion. We show that the method provides consistent and convergent approximations of surface vector fields and differential operators. We study the relationship between error, spatial resolution, and artificial Mach number and characterize the frequency spectrum of the artificial oscillations. We provide numerical solutions of the incompressible Navier-Stokes equations on symmetric surfaces, such as the sphere and torus, and on parametric and non-parametric asymmetric surfaces. Since the proposed method works directly on unstructured surface point clouds, it provides a promising approach for simulations on image-derived geometries, such as in biological morphogenesis from microscopy videos.

A Cartesian Grid Method for Advection-Diffusion Equations with Robin Boundary Conditions on Moving Domains

We develop a Cartesian grid method for advection--diffusion equations with Robin boundary conditions on moving domains. The moving-domain problem is reformulated as an interface problem on a box, with an unknown density introduced on the moving interface to enforce the Robin condition. The bulk equation is discretized by a cell-centered finite-difference scheme on the Cartesian grid, while interface corrections are obtained from local problems in a narrow band around the interface. The resulting method requires only modest computational geometry, avoids remeshing and cut cells, and is compatible with geometric multigrid and matrix-free GMRES. The GMRES iteration count is essentially independent of the mesh size, and the computational cost scales linearly with the number of bulk degrees of freedom. For the one-dimensional scheme, first-order convergence in time and second-order convergence in space are proved. Numerical examples in one and two dimensions, including manufactured solutions and an active transport problem without an exact solution, demonstrate the accuracy and efficiency of the method.

Validation of a 3-D high-order discontinuous Galerkin solver with transducer boundary conditions through comparisons with the fast nearfield method

A 3-D discontinuous Galerkin (DG) solver is presented for ultrasound propagation in heterogeneous media. The formulation advances particle velocity and density perturbation fields using explicit time integration, while numerical fluxes couple elements and enforce material interfaces, enabling layered configurations and spatially varying material properties. DG is ideal for these calculations due to high-order accuracy, element-local operators, and a flux-based boundary-condition model for finite-aperture sources. Implementation choices governing accuracy and computational cost are considered, comparing consistent and diagonal (lumped) formulations. Transducer boundary conditions are implemented for single-element transducers and linear arrays. Two enforcement strategies are investigated: a pointwise imposition of the boundary drive at face evaluation points, and an L2 projection of the drive onto the face polynomial space. The projected approach reduces aliasing for coarse meshes and high-order discretizations by representing boundary data consistently within the face polynomial basis. Linear validation in homogeneous media is performed via comparisons to the fast nearfield method for single-element and multi-element configurations, including focused sub-apertures with transmit delays. Transient pressure fields and centerline waveforms are compared in order to demonstrate accurate boundary-driven excitation across various aperture and focusing conditions.