There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa

AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20


Published: SIAM Review, 44 (2002) 631--658.

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Convergence of Adaptive Finite Element Methods

There is an abundancy of systems characterized by parabolic PDEs in science and engineering, especially in chemistry and physics. These systems have a scalar variable, we generally call time, defining the evolution of the system under consideration. The governing equation(s) involves the unknown(s) and their first order partial derivative(s) with respect to this variable. Time variant Schrodinger equations where the unknown is the wavefunction which is responsible for the probability density for the system and Liouville equations for the statistical mechanics where the unknown is somehow responsible for a density in the systems’ phase space (here we use the plurality since both case may differ from Hamiltonian to Hamiltonian). Certain PDE(s), depending on so-called spatial coordinates, govern the behavior of the system in these and similar cases even though the partial differential equation nature is not necessarily needed. Hence we give the following equation for more

MATHEMATICAL MODELS and METHODS in APPLIED SCIENCES

http://dilbert.engr.ucdavis.edu/~suku/quadrature/xfemquadrature.pdf

Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method

A new partition of unity finite element free from the linear dependence problem and possessing the delta property

Meshfree particle methods for thin plates

Mapping Techniques for Isogeometric Analysis of Elliptic Boundary Value Problems Containing Singularities

In the previous papers (Kim et al. Submitted for publication, Oh et al. in press), for uniformly or locally non-uniformly distributed particles, we constructed highly regular piecewise polynomial particle shape functions that have the polynomial reproducing property of order k for any given integer k ≥ 0 and satisfy the Kronecker Delta Property. In this paper, in order to make these particle shape functions more useful in dealing with problems on complex geometries, we introduce smooth-piecewise-polynomial Reproducing Polynomial Particle shape functions, corresponding to the particles that are patch-wise non-uniformly distributed in a polygonal domain. In order to make these shape functions with compact supports, smooth flat-top partition of unity shape functions are constructed and multiplied to the shape functions. An error estimate of the interpolation associated with such flexible piecewise polynomial particle shape functions is proven. The one-dimensional and the two-dimensional numerical results that support the theory are resented.

/pdf/the-smooth-piecewise-polynomial-particle-shape-functions-2afs8tijp5.pdf

The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particle methods

SUMMARY

The mapping method was introduced in Jeong et al. (2013) for highly accurate isogeometric analysis (IGA) of elliptic boundary value problems containing singularities. The mapping method is concerned with constructions of novel geometrical mappings by which push-forwards of B-splines from the parameter space into the physical space generate singular functions that resemble the singularities. In other words, the pullback of the singularity into the parameter space by the novel geometrical mapping (a non-uniform rational basis spline (NURBS) surface mapping) becomes highly smooth. One of the merits of IGA is that it uses NURBS functions employed in designs for the finite element analysis. However, push-forwards of rational NURBS may not be able to generate singular functions. Moreover, the mapping method is effective for neither the k-refinement nor the h-refinement. In this paper, highly accurate stress analysis of elastic domains with cracks and ∕ or corners are achieved by enriched IGA, in which push-forwards of NURBS via the design mapping are combined with push-forwards of B-splines via the novel geometrical mapping (the mapping technique). In a similar spirit of X-FEM (or GFEM), we propose three enrichment approaches: enriched IGA for corners, enriched IGA for cracks, and partition of unity IGA for cracks. Copyright © 2013 John Wiley & Sons, Ltd.

Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

In this paper, we construct particle shape functions that reproduce singular functions as well as polynomial functions. We also construct piecewise polynomial convolution partition of unity functions by taking the convolution of the scaled conical window function with the characteristic functions of quadrangular patches (we provide the computer code for this construction). We demonstrate that the reproducing singular particle shape functions yield highly accurate numerical solutions for the singularity problems with crack singularity or a jump boundary data singularity.

/pdf/the-reproducing-singularity-particle-shape-functions-for-1ebiopnbln.pdf

Jae Woo Jeong

Papers

Meshfree particle methods for thin plates

Mapping Techniques for Isogeometric Analysis of Elliptic Boundary Value Problems Containing Singularities

The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particle methods

Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

The reproducing singularity particle shape functions for problems containing singularities