An adaptive multilevel Monte Carlo algorithm for the stochastic drift-diffusion-Poisson system. (English) Zbl 1462.65166
Summary: We present an adaptive multilevel Monte Carlo algorithm for solving the stochastic drift-diffusion-Poisson system with non-zero recombination rate. The a-posteriori error is estimated to enable goal-oriented adaptive mesh refinement for the spatial dimensions, while the a-priori error is estimated to guarantee linear convergence of the \(H^1\) error. In the adaptive mesh refinement, efficient estimation of the error indicator gives rise to better error control. For the stochastic dimensions, we use the multilevel Monte Carlo method to solve this system of stochastic partial differential equations. Finally, the advantage of the technique developed here compared to uniform mesh refinement is discussed using a realistic numerical example.
MSC:
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 82D37 | Statistical mechanics of semiconductors |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
Keywords:
stochastic partial differential equation (SPDEs); stochastic drift-diffusion-Poisson system; adaptive mesh refinement; a priori error estimation; a posteriori error estimation; multilevel Monte CarloSoftware:
ALEAReferences:
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