Stochastic variational multiscale method for error estimation and adaptivity in uncertain transport problems

Abstract

Similarly, a model term is derived to explicitly estimate the error in a local or element-wise fashion. This model term is approximated using the components of the stabilization parameter used in computing the numerical solution, making error estimation computationally inexpensive. We compare the error estimator with either the true error or a reference error from a much finer discretization, with which our error estimator agrees very well both locally and globally. Further, procedures using the local error estimator are designed to drive adaptivity in the physical domain and in the stochastic domain. In the physical domain, we apply mesh adaptation. Likewise, stochastic adaptivity controls the local spectral approximation (i.e., a spatially varying spectral order over the mesh). We propose two schemes for adaptivity and apply them to mesh adaptivity and stochastic adaptivity individually. We demonstrate adaptivity on several transport problems with up to three orders of savings in the number of degrees-of-freedom for a given level of accuracy.

The focus of this work is the formulation and application of an adaptive approach based on the variational multiscale (VMS) method for stochastic PDEs with uncertain input data. Uncertainty leads to complicated solution behavior and features in both the physical and stochastic domains. These features can be local, requiring high resolution in some portions to be accurate, whereas in other portions, a low level of resolution is sufficient. For such problems, we seek adaptive construction of efficient discretizations which selectively have high resolution in portions with complicated solution behavior.

In this approach, we employ finite elements in the spatial domain and spectral approximation (based on generalized polynomial chaos) in the stochastic domain. The stochastic VMS method allows in computing an accurate solution while accounting for the missing or fine scales through a model term. This model term is algebraically approximated in each element using a stochastic stabilization parameter. We demonstrate that our stochastic VMS methodology provides a stable and accurate solution for complex transport problems.