Sparse Polynomial Approximations for Affine Parametric Saddle Point Problems

In this work we study convergence properties of sparse polynomial approximations for a class affine parametric saddle point problems, including the Stokes equations viscous incompressible flow, mixed formulation diffusion groundwater time-harmonic Maxwell electromagnetics, etc. Due to lack knowledge or intrinsic randomness, (viscosity, diffusivity, permeability, permittivity, etc.) coefficients such problems are uncertain and can often be represented approximated by high- countably infinite-dimensional random parameters equipped with suitable probability distributions, affinely depend on series either globally locally supported basis functions, e.g., Karhunen–Loève expansion, piecewise polynomials, adaptive wavelet approximations. We consider solutions, in particular Taylor approximations, their these problems. Under sparsity assumptions parametrization coefficients, show algebraic rates O(N−r) solutions based results elliptic PDEs (Cohen, A. et al.: Anal. Appl. 9, 11–47, 2011), (Bachmayr, M., ESAIM Math. Model. Numer. 51, 321–339, 2017), A., DeVore, R.: Acta 24, 1–159, 2015), (Chkifa, J. Pures 103, 400–428, 47, 253–280, 2013), Migliorati, G.: Contemp. Comput. Math., 233–282, 2018), rate r depending only parameter parametrization, not number active dimensions terms N. note that were considered 2015, Section 2.2) anticipation same approximation solution map obtained extended more general paper, different from presented 2.2), obtain two variables, velocity pressure equations, which case functions.