On the multi-level splitting of finite element spaces
Numerische Mathematik
Special finite element methods for a class of second order elliptic problems with rough coefficients
SIAM Journal on Numerical Analysis
Journal of Computational Physics
High accuracy solution of Maxwell's equations using nonstandard finite differences
Computers in Physics
Efficient pseudospectral flow simulations in moderately complex geometries
Journal of Computational Physics
A discontinuous hp finite element method for diffusion problems
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A short survey on preconditioning techniques in spectral calculations
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs
SIAM Journal on Scientific Computing
A Particle-Partition of Unity Method--Part III: A Multilevel Solver
SIAM Journal on Scientific Computing
A Particle-Partition of Unity Method--Part II: Efficient Cover Construction and Reliable Integration
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
SIAM Journal on Numerical Analysis
A new finite difference method for a class of singular two-point boundary value problems
Applied Mathematics and Computation
Discontinuous Galerkin Methods: Theory, Computation and Applications
Discontinuous Galerkin Methods: Theory, Computation and Applications
Optimized three-dimensional FDTD discretizations of Maxwell's equations on Cartesian grids
Journal of Computational Physics
Trefftz difference schemes on irregular stencils
Journal of Computational Physics
A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates
Journal of Scientific Computing
Parallel solvers for flexible approximation schemes in multiparticle simulation
ICCS'06 Proceedings of the 6th international conference on Computational Science - Volume Part I
Fourth-order finite-difference time-domain method based on error-controlling concepts
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields
Hi-index | 31.46 |
Solutions of many physical problems have salient local features that are qualitatively known a priori (for example, singularities at point sources, edge and corners; boundary layers; derivative jumps at material interfaces; strong dipole field components near polarized spherical particles; cusps of electronic wavefunctions at the nuclei; electrostatic double layers around colloidal particles, etc.) The known methods capable of providing flexible local approximation of such features include the generalized finite element - partition of unity method, special variational-difference schemes in broken Sobolev spaces, and a few other specialized techniques. In the proposed new class of Flexible Local Approximation MEthods (FLAME), a desirable set of local approximating functions (such as cylindrical or spherical harmonics, plane waves, harmonic polynomials, etc.) defines a finite difference scheme on a chosen grid stencil. One motivation is to minimize the notorious 'staircase' effect at curved and slanted interface boundaries. However, the new approach has much broader applications. As illustrative examples, the paper presents arbitrarily high order 3-point schemes for the 1D Schrodinger equation and a 1D singular equation, schemes for electrostatic interactions of colloidal particles, electromagnetic wave propagation and scattering, plasmon resonances. Moreover, many classical finite difference schemes, including the Collatz ''Mehrstellen'' schemes, are direct particular cases of FLAME.