Adjusted sparse tensor product spectral Galerkin method for solving pseudodifferential equations on the sphere with random input data. (English) Zbl 1435.65209
Summary: An adjusted sparse tensor product spectral Galerkin approximation method based on spherical harmonics is introduced and analyzed for solving pseudodifferential equations on the sphere with random input data. These equations arise from geodesy where the sphere is taken as a model of the earth. Numerical solutions to the corresponding \(k\)-th order statistical moment equations are found in adjusted sparse tensor approximation spaces which are accordingly designed to the regularity of the data and the equation. Established convergence theorem shows that the adjusted sparse tensor Galerkin discretization is superior not only to the full tensor product but also to the standard sparse tensor counterpart when the statistical moments of the data are of mixed unequal regularity. Numerical experiments illustrate our theoretical results.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 41A25 | Rate of convergence, degree of approximation |
| 35S15 | Boundary value problems for PDEs with pseudodifferential operators |
| 86A30 | Geodesy, mapping problems |
Keywords:
stochastic pseudodifferential equations; statistical moments; hyperbolic cross spectral methods; spheresReferences:
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