High dimensional finite elements for time-space multiscale parabolic equations
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High dimensional finite elements for time-space multiscale parabolic equations Wee Chin Tan1 · Viet Ha Hoang1 Received: 5 January 2018 / Accepted: 18 December 2018 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract The paper develops the essentially optimal sparse tensor product finite element method for a parabolic equation in a domain in Rd which depends on a microscopic scale in space and a microscopic scale in time. We consider the critical self similar case which has the most interesting homogenization limit. We solve the high dimensional time-space multiscale homogenized equation, which provides the solution to the homogenized equation which describes the multiscale equation macroscopically, and the corrector which encodes the microscopic information. For obtaining an approximation within a prescribed accuracy, the method requires an essentially optimal number of degrees of freedom that is essentially equal to that for solving a macroscopic parabolic equation in a domain in Rd . A numerical corrector is deduced from the finite element solution. Numerical examples for one and two dimensional problems confirm the theoretical results. Although the theory is developed for problems with one spatial microscopic scale, we show numerically that the method is capable of solving problems with more than one spatial microscopic scale. Keywords High dimensional finite elements · Time-space multiscale parabolic equations · Optimal complexity · Numerical corrector Mathematics Subject Classification (2010) 35B27 · 65M12 · 65M60
1 Introduction Let D ⊂ Rd be a bounded domain where d = 1, 2 or 3. Let T > 0. Let Y = (0, 1)d be the unite cube in Rd . We consider a symmetric matrix valued function Communicated by: Ivan Oseledets Viet Ha Hoang
[email protected] 1
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
W.C. Tan, V.H. Hoang
a(t, x, τ, y) ∈ C([0, T ] × D¯ × [0, 1] × Y¯ ; Rd×d sym ). The function a is periodic with respect to τ and y, with the period being (0, 1) and Y respectively; from now on, we say that it is (0, 1) × Y periodic with respect to τ and y. We assume that a is uniformly coercive and bounded, i.e, there are positive constants c1 and c2 such that for all ξ, ζ ∈ Rd c1 |ξ |2 ≤ a(t, x, τ, y)ξ · ξ, a(t, x, τ, y)ξ · ζ ≤ c1 |ξ ||ζ |
(1.1)
for all (t, x, τ, y) ∈ (0, T ) × D × (0, 1) × Y , where | · | denotes the Euclidean norm in Rd . Let ε > 0 be a small number that represents the microscopic scale. We consider the time-space multiscale coefficient t x ε a (t, x) = a t, x, 2 , . ε ε We denote by V = H01 (D) and H = L2 (D). We note that V ⊂ H ⊂ V form a Gelfand triple. We define by ·, ·H the inner product in H, extended to the duality pairing between V and V by density. Let T > 0, f ∈ L2 ((0, T ), V ) and g ∈ H . We consider the parabolic problem ∂uε − ∇ · (a ε (t, x)∇uε ) = f (t, x), x ∈ D, t ∈ (0, T ), ∂t uε (0, x) = g, x ∈ D
(1.2)
with the Dirichlet boundary condition fo
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