Sparse finite element approximation of high-dimensional transport-dominated diffusion problems
Seminar für Angewandte Mathematik, Eidgenössische Technische Hochschule,
8092 Zürich, Switzerland. firstname.lastname@example.org; email@example.com
2 University of Oxford, Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. firstname.lastname@example.org
We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form , , where is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation uh on a partition of Ω of mesh size h = hL = 2-L satisfies the following bound in the streamline-diffusion norm , provided u belongs to the space of functions with square-integrable mixed (k+1)st derivatives: where , i=0,1, and . We show, under various mild conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion problems , and hence for p ≥ 1 the 'error constant' exhibits exponential decay as d → ∞; in the case of a general symmetric positive semidefinite matrix a, the error constant is shown to grow no faster than . In any case, in the absence of assumptions that relate L, p and d, the error is still bounded by , where for all L, p, d ≥ 2.
Mathematics Subject Classification: 65N30
Key words: High-dimensional Fokker-Planck equations / partial differential equations with nonnegative characteristic form / sparse finite element method.
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