Math 2.0

Numerical analysis Definition

Meshfree methods

Builds on Moving least squaresRadial basis functionPartition of unityGalerkin method

Definition

Let Ω⊂ℝd\Omega \subset \R^d be a domain and let X={x1,…,xn}⊂Ω‾X = \set{x_1, \ldots, x_n} \subset \overline{\Omega} be a finite set of nodes. A meshfree method for a differential equation on Ω\Omega approximates the solution by uh(x)=∑i=1nφi(x) ci,u_h(x) = \sum_{i=1}^{n} \varphi_i(x) \, c_i, where the functions φi\varphi_i and the method satisfy two conditions:

  1. each φi\varphi_i is determined by the node set XX and fixed weight or kernel functions alone, with no elements or edges between nodes;
  2. the coefficients cic_i are found by a Galerkin condition, by collocation at points of Ω\Omega, or by a particle evolution law.

Usually each φi\varphi_i has compact support, a ball of radius ρ\rho around xix_i.

Remarks

Common choices of φi\varphi_i are moving least squares shape functions and radial basis functions φi(x)=ϕ(∣x−xi∣)\varphi_i(x) = \phi(\abs{x - x_i}). Moving least squares shape functions of order mm reproduce polynomials of degree at most mm; since they reproduce constants, they form a partition of unity. Essential boundary conditions need extra care, because these φi\varphi_i usually do not satisfy φi(xj)=δij\varphi_i(x_j) = \delta_{ij}.

Example: Gaussian radial basis function collocation for −Δu=f-\Delta u = f uses φi(x)=e−ε2∣x−xi∣2\varphi_i(x) = e^{-\varepsilon^2 \abs{x - x_i}^2} with a shape parameter ε>0\varepsilon > 0. These functions depend only on the nodes and on ε\varepsilon, which satisfies clause 1. The cic_i solve −Δuh(xj)=f(xj)-\Delta u_h(x_j) = f(x_j) at interior nodes and the boundary condition at boundary nodes, which is clause 2.

Non-example: the hat functions of linear finite elements on a triangulation are not meshfree. Each hat function depends on which triangles contain its node, so clause 1 fails.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Galerkin method, concept, no evidence levelGalerkin methodconceptMoving least squares, concept, no evidence levelMoving least squaresconceptPartition of unity, concept, no evidence levelPartition of unityconceptRadial basis function, concept, no evidence levelRadial basis functionconceptMeshfree methods, definition, no evidence levelMeshfree methodsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Galerkin method concept
  2. Moving least squares concept
  3. Partition of unity concept
  4. Radial basis function concept
  5. Meshfree methods definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Meshfree_methods
About this page
Tag30GF7TK8YP
URIm2://30GF7TK8YP
Kinddefinition
WritingW0 stub
MSC65N99, 65M99, 65D15
DefinesMeshfree methods
UsesMoving least squares, Radial basis function, Partition of unity, Galerkin method

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