Math 2.0

Numerical analysis Definition

Method of fundamental solutions

Builds on Fundamental solutionLaplacianDirichlet problem

Definition

Let Ω⊂ℝd\Omega \subset \R^d be a bounded domain and LL a linear elliptic differential operator with constant coefficients. Let GG be a fundamental solution of LL, so that LG=δL G = \delta; it is smooth on ℝd∖{0}\R^d \setminus \set{0}. Consider Lu=0L u = 0 in Ω\Omega with the condition u=gu = g on ∂Ω\partial \Omega. The method of fundamental solutions has three ingredients:

  1. source points y1,…,yny_1, \ldots, y_n in ℝd∖Ω‾\R^d \setminus \overline{\Omega};
  2. the trial function un(x)=∑j=1ncj G(x−yj)u_n(x) = \sum_{j=1}^{n} c_j \, G(x - y_j), which satisfies Lun=0L u_n = 0 in Ω\Omega;
  3. boundary points x1,…,xm∈∂Ωx_1, \ldots, x_m \in \partial \Omega, with m≥nm \ge n.

The coefficients cjc_j solve un(xi)=g(xi)u_n(x_i) = g(x_i) for all ii, exactly if m=nm = n and in the least squares sense if m>nm > n.

Remarks

The sources lie outside Ω‾\overline{\Omega}, so each G(⋅−yj)G(\cdot - y_j) is smooth on Ω‾\overline{\Omega} and solves the equation there. The error comes only from the boundary condition. The sources are often placed on a curve or surface surrounding Ω\Omega at some distance. The linear system becomes very ill-conditioned as nn grows, which limits the accuracy reached in floating point arithmetic.

Example: the Dirichlet problem for the Laplace equation Δu=0\Delta u = 0, with Δ\Delta the Laplacian, in the unit disk of ℝ2\R^2. Take L=−ΔL = -\Delta, with fundamental solution G(x)=−12πln⁡∣x∣G(x) = -\frac{1}{2\pi} \ln \abs{x}. Take sources yj=R(cos⁡θj,sin⁡θj)y_j = R (\cos \theta_j, \sin \theta_j) with θj=2πj/n\theta_j = 2\pi j / n and R>1R > 1. Take as many boundary points xk=(cos⁡αk,sin⁡αk)x_k = (\cos \alpha_k, \sin \alpha_k), with αk=2π(k−12)/n\alpha_k = 2\pi (k - \tfrac12) / n. Every unu_n is harmonic in the disk, and the cjc_j are fixed by matching gg at the xkx_k.

Non-example: with a source point inside Ω\Omega, clause 1 fails. Then unu_n has a logarithmic singularity in Ω\Omega and does not solve the equation there.

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Boundary, concept, no evidence levelBoundaryconceptContinuous function, concept, no evidence levelContinuous functionconceptDomain in the complex plane, concept, no evidence levelDomain in the complex…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFundamental solution, concept, no evidence levelFundamental solutionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptHarmonic function, concept, no evidence levelHarmonic functionconceptDirichlet problem, problem, no evidence levelDirichlet problemproblemMethod of fundamental solutions, definition, no evidence levelMethod of fundamental…definition
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  1. Boundary concept
  2. Continuous function concept
  3. Domain in the complex plane concept
  4. Euclidean space object
  5. Function concept
  6. Fundamental solution concept
  7. Total derivative concept
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Differentiable function of several variables concept
  11. Partial derivative concept
  12. Vector field concept
  13. Gradient concept
  14. Divergence concept
  15. Laplacian concept
  16. Harmonic function concept
  17. Dirichlet problem problem
  18. Method of fundamental solutions definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Method_of_fundamental_solutions
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MSC65N80, 65N35
DefinesMethod of fundamental solutions
UsesFundamental solution, Laplacian, Dirichlet problem

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