High dimensional finite elements for time-space multiscale parabolic equations

Abstract

The paper develops the essentially optimal sparse tensor product finite element method for a parabolic equation in a domain in $\mathbb {R}^{d}$ 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 $\mathbb {R}^{d}$. 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.

Appendix

We prove Theorem 3.2 in this appendix. Let $\rho _{0,m}={1\over {\Delta } t}(u_{0}(t_{m + 1})-u_{0}(t_{m}))-{\partial u_{0}\over \partial t}(t_{m + 1/2})$, $\zeta _{0,m}=\frac 12(u_{0}(t_{m + 1})+u_{0}(t_{m}))-u_{0}(t_{m + 1/2})$, $\zeta _{1,m}=\frac 12(u_{1}(t_{m + 1})+u_{1}(t_{m}))-u_{1}(t_{m + 1/2})$ and $\xi _{1,m}={1\over 2}\left ({\partial u_{1}\over \partial \tau }(t_{m + 1})+{\partial u_{1}\over \partial \tau }(t_{m})\right )-{\partial u_{1}\over \partial \tau }(t_{m + 1/2})$. Since u₀ ∈ C³([0, T], H) ∩ C²([0, T], V ), u₁ ∈ C²([0, T], L²(D × (0, 1), V_#)) and ${\partial u_{1}\over \partial \tau }\in C^{2}([0,T],L^{2}(D\times (0,1),V^{\prime }_\#))$, we deduce that

$$\begin{array}{@{}rcl@{}} && \|\rho_{0,m}\|_{L^{2}(D)}\le c({\Delta} t)^{2}, \|\zeta_{0,m}\|_{V}\le c({\Delta} t)^{2}, \|\zeta_{1,m}\|_{V_{1}}\le c({\Delta} t)^{2}, \text{ and }\\ &&\|\xi_{1,m}\|_{L^{2}(D\times(0,1),V_{\#}^{\prime})}\le c({\Delta} t)^{2} \end{array} $$

where the constant c does not depend on m. From (2.1) and (3.1) considered at t = t_m+ 1/2 we deduce that

$$\begin{array}{@{}rcl@{}} &&\left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},\phi_{0}\right\rangle_{H}+\langle\rho_{0,m},\phi_{0}\rangle_{H}\\ &&\qquad +{\int}_{D}{{\int}_{0}^{1}}\left\langle{1\over 2}\left( {\partial z_{1,m + 1}\over\partial\tau}+{\partial z_{1,m}\over\partial\tau}\right),\phi_{1}\right\rangle_{H_\#} d\tau dx+ {\int}_{D}{{\int}_{0}^{1}}\langle\xi_{1},\phi_{1}\rangle_{H_\#} d\tau dx\\ &&\qquad +{\int}_{D}{{\int}_{0}^{1}}{\int}_{Y}a(t_{m + 1/2})\left( \nabla_{x}{z_{0,m + 1}+z_{0,m}\over 2}+\nabla_{y}{z_{1,m + 1}+z_{1,m}\over 2}\right)\\ &&\qquad \cdot(\nabla_{x}\phi_{0}+\nabla_{y}\phi_{1})dyd\tau dx\\ &&\qquad+{\int}_{D}{{\int}_{0}^{1}}{\int}_{Y}a(t_{m + 1/2})(\nabla_{x}\zeta_{0,m}+\nabla_{y}\zeta_{1,m})\cdot(\nabla_{x}\phi_{0}+\nabla_{y}\phi_{1})dyd\tau dx= 0.\\ \end{array} $$

(A.1)

Consider

$$\begin{array}{@{}rcl@{}} I &=& \left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},{z_{0,m + 1}+z_{0,m}\over2}\right\rangle_{H}\\ &&+ {\int}_{D}{{\int}_{0}^{1}}\left\langle{1\over 2}\left( {\partial z_{1,m + 1}\over\partial\tau}+{\partial z_{1,m}\over\partial\tau}\right),{z_{1,m + 1}+z_{1,m}\over 2}\right\rangle_{H_\#}d{\tau}dx\\ &&+{\int}_{D}{{\int}_{0}^{1}}{\int}_{Y}a(t_{m + 1/2})\left( \nabla_{x}{z_{0,m + 1}+z_{0,m}\over2}+\nabla_{y}{z_{1,m + 1}+z_{1,m}\over 2}\right)\\ && \cdot \left( \nabla_{x}{z_{0,m + 1}+z_{0,m}\over2}+\nabla_{y}{z_{1,m + 1}+z_{1,m}\over 2}\right)dyd{\tau}dx\\ &\ge& {1\over2{\Delta} t}(\|z_{0,m + 1}\|^{2}_{H}-\|z_{0,m}\|_{H}^{2})+\gamma(\|z_{0,m + 1/2}\|^{2}_{V}+\|z_{1,m + 1/2}\|^{2}_{V_{1}}).\qquad \end{array} $$

(A.2)

For $\{\tilde u_{0,m},\ m = 0,\ldots ,M\}\subset V^{L}$ and $\{\tilde u_{1,m},\ m = 1,\ldots ,M\}\subset {V_{1}^{L}}$, we have

$$\begin{array}{@{}rcl@{}} I&=&\left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},(u_0-\tilde u_0)_{m + 1/2}\right\rangle_H+\left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},(\tilde u_0-U_0)_{m + 1/2}\right\rangle_H\\ &&+{\int}_D{\int}_0^{1}\left\langle{1\over2}\left( {\partial z_{1,m + 1}\over\partial\tau}+{\partial z_{1,m}\over\partial\tau}\right),(u_1-\tilde u_1)_{m + 1/2}\right\rangle_{H_\#}d\tau dx\\ &&+{\int}_D{\int}_0^{1}\left\langle{1\over2}\left( {\partial z_{1,m + 1}\over\partial\tau}+{\partial z_{1,m}\over\partial\tau}\right),(\tilde u_1-U_1)_{m + 1/2}\right\rangle_{H_\#}d\tau dx\\ &&+{\int}_D{\int}_0^{1}{\int}_Ya(t_{m + 1/2})\left( \nabla_x{z_{0,m + 1}+z_{0,m}\over 2}+\nabla_y{z_{1,m + 1}+z_{1,m}\over 2}\right)\\ &&\cdot\left( \nabla_x(u_0-\tilde u_0)_{m + 1/2}+\nabla_y(u_1-\tilde u_1)_{m + 1/2}\right)dyd\tau dx\\ &&+{\int}_D{\int}_0^{1}{\int}_Ya(t_{m + 1/2})\left( \nabla_x{z_{0,m + 1}+z_{0,m}\over 2}+\nabla_y{z_{1,m + 1}+z_{1,m}\over 2}\right)\\ &&\cdot \left( \nabla_x(\tilde u_0-U_0)_{m + 1/2}+\nabla_y(\tilde u_1-U_1)_{m + 1/2}\right)dyd\tau dx. \end{array} $$

From (3.1) we have

$$\begin{array}{@{}rcl@{}} I&=&\left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},(u_0-\tilde u_0)_{m + 1/2}\right\rangle_H\\ &&-{\int}_D{\int}_0^{1}\left\langle{z_{1,m + 1}+z_{1,m}\over2},{\partial\over\partial\tau}(u_1-\tilde u_1)_{m + 1/2}\right\rangle_{H_\#}d{\tau}dx\\ &&+{\int}_D{\int}_0^{1}{\int}_Ya(t_{m + 1/2})\left( \nabla_x{z_{0,m + 1}+z_{0,m}\over 2}+\nabla_y{z_{1,m + 1}+z_{1,m}\over 2}\right)\\ &&\cdot\left( \nabla_x(u_0-\tilde u_0)_{m + 1/2}+\nabla_y(u_1-\tilde u_1)_{m + 1/2}\right)dyd\tau dx\\ &&- \langle\rho_{0,m},(\tilde u_0-U_0)_{m + 1/2}\rangle_H-{\int}_D{\int}_0^{1}\langle\xi_{1,m},(\tilde u_1-U_1)_{m + 1/2}\rangle_{H_\#}d\tau dx\\ &&+{\int}_D{\int}_0^{1}{\int}_Ya(t_{m + 1/2})\left( \nabla_x\zeta_{0,m}+\nabla_y\zeta_{1,m}\right)\\ &&\cdot\left( \nabla_x(\tilde u_0-U_0)_{m + 1/2}+\nabla_y(\tilde u_1-U_1)_{m + 1/2}\right)dyd\tau dx. \end{array} $$

We note that $(\tilde u_{0}-U_{0})_{m + 1/2}=(\tilde u_{0}-u_{0})_{m + 1/2}+z_{0,m + 1/2}$ and $(\tilde u_{1}-U_{1})_{m + 1/2}=(\tilde u_{1}-u_{1})_{m + 1/2}+z_{1,m + 1/2}$. For a positive constant δ > 0, using the Cauchy-Schwartz inequality, we have

$$\begin{array}{@{}rcl@{}} I\!\!&\le&\!\left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},(u_0-\tilde u_0)_{m + 1/2}\right\rangle_H\\ &&\!+ \delta\|z_{1,m + 1/2}\|_{V_1}^{2}+c\left\|{\partial\over\partial\tau}(u_1-\tilde u_1)_{m + 1/2}\right\|_{V_{1}^{\prime}}^{2}\\ &&\!+\delta\|z_{0,m + 1/2}\|^{2}_V+\delta\|z_{1,m + 1/2}\|^{2}_{V_1} +c\|(u_0-\tilde u_0)_{m + 1/2}\|_V^{2}+c\|(u_1-\tilde u_1)_{m + 1/2}\|_{V_1}^{2}\\ &&+c\|\rho_{0,m}\|_H^{2}+c\|(\tilde u_0-u_0)_{m + 1/2}\|^{2}_H+\delta\|z_{0,m + 1/2}\|_H^{2}\\ &&+c\|\xi_{1,m}\|^{2}_{V_{1}^{\prime}}+c\|(\tilde u_1-u_1)_{m + 1/2}\|^{2}_{V_1}+\delta\|z_{1,m + 1/2}\|^{2}_{V_1}\\ &&+c\|\zeta_{0,m}\|_V^{2}+c\|\zeta_{1,m}\|_{V_1}^{2} +c\|(\tilde u_0-u_0)_{m + 1/2}\|^{2}_V+\delta\|z_{0,m + 1/2}\|_V^{2}\\ &&+c\|(\tilde u_1-u_1)_{m + 1/2}\|^{2}_{V_1}+\delta\|z_{1,m + 1/2}\|_V^{2}. \end{array} $$

From this and (A.2), choosing δ sufficiently small, we have

$$\begin{array}{@{}rcl@{}} &&{}{1\over 2{\Delta} t}(\|z_{0,m + 1}\|^{2}_H-\|z_{0,m}\|_H^{2})+c(\|z_{0,m + 1/2}\|^{2}_V+\|z_{1,m + 1/2}\|^{2}_{V_1})\\ &&\quad{} \le\!\left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},(u_0-\tilde u_0)_{m + 1/2}\right\rangle_H\\ &&\qquad{}\! +\! c\left\|{\partial\over\partial\tau}(u_1 - \tilde u_1)_{m + 1{\kern-.5pt}/{\kern-.5pt}2}\right\|_{V_{1}^{\prime}}^{2}+c\|{\kern-.5pt}({\kern-.5pt}u_0-\tilde u_0{\kern-.5pt})_{m{\kern-.5pt}+{\kern-.5pt}1{\kern-.5pt}/2}\|_V^{2} + c{\kern-.5pt}\|{\kern-.5pt}({\kern-.5pt}u_1-\tilde u_1{\kern-.5pt})_{m{\kern-.5pt}+{\kern-.5pt}1{\kern-.5pt}/{\kern-.5pt}2}\|_{{\kern-.5pt}V_1}^{2}\!{\kern-.5pt}+\!c({\kern-.5pt}{\Delta} t{\kern-.5pt})^4{\kern-.5pt}. \end{array} $$

Fixing an integer P ≤ M, taking the sum for m = 0, … , P − 1, we have

$$\begin{array}{@{}rcl@{}} &&\|z_{0,P}\|_{H}^{2}-\|z_{0,0}\|_{H}^{2}+c{\Delta} t\sum\limits_{m = 0}^{P-1}(\|z_{0,m + 1/2}\|_{V}^{2}+\|z_{1,m + 1/2}\|^{2}_{V_{1}})\\ &&\le c{\Delta} t\sum\limits_{m = 0}^{P-1}\left( \left\|{\partial\over\partial\tau}(u_{1}-\tilde u_{1})_{m + 1/2}\right\|_{V_{1}^{\prime}}^{2} + \|(u_{0}-\tilde u_{0})_{m + 1/2}\|_{V}^{2}+\|(u_{1}-\tilde u_{1})_{m + 1/2}\|_{V_{1}}^{2}\right)\\ &&cP({\Delta} t)^{5}+ 2{\Delta} t\sum\limits_{m = 0}^{P-1}\left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},(u_{0}-\tilde u_{0})_{m + 1/2}\right\rangle_{H}. \end{array} $$

(A.3)

We note that

$$\begin{array}{@{}rcl@{}} &&{\Delta} t{\sum}_{m = 0}^{P-1}\left\langle{z_{0,m + 1}-z_{0,m}\over{\Delta} t},(u_0-\tilde u_0)_{m + 1/2}\right\rangle_H\\ && \qquad\qquad=\left\langle z_{0,P},(u_0-\tilde u_0)_{P-1/2}\right\rangle_H-\left\langle z_{0,0},(u_0-\tilde u_0)_{1/2}\right\rangle_H\\ &&\qquad\qquad\qquad+{\Delta} t{\sum}_{m = 1}^{P-1}\left\langle{z_{0,m}\over{\Delta} t},(u_0-\tilde u_0)_{m-1/2}-(u_0-\tilde u_0)_{m + 1/2}\right\rangle_H\\ &&\qquad\qquad\le \delta\|z_{0,P}\|_H^{2}+c\|(u_0-\tilde u_0)_{P-1/2}\|_H^{2}+\|z_{0,0}\|_H^{2}+\|(u_0-\tilde u_0)_{1/2}\|_H^{2}\\ &&\qquad\qquad\qquad+\delta{\Delta} t{\sum}_{m = 1}^{P-1}\|z_{0,m}\|_H^{2}+c{\Delta} t{\sum}_{m = 1}^{P-1}\left\|{(u_0-\tilde u_0)_{m + 1/2}-(u_0-\tilde u_0)_{m-1/2}\over{\Delta} t}\right\|_H^{2} \end{array} $$

which is a consequence of the Cauchy-Schwartz inequality; δ is an arbitrary positive constant. We note that ${\Delta } t{\sum }_{m = 1}^{P-1}\|z_{0,m}\|_{H}^{2}\le {\Delta } t P\max _{m = 0,\ldots ,M}\|z_{0,m}\|_{H}^{2}\le T\max _{m = 0,\ldots ,M}\|z_{0,m}\|_{H}^{2}$. From this and (A.3), choosing δ sufficiently small, we have

$$\begin{array}{@{}rcl@{}} \|z_{0,P}\|^{2}_H\!&\le&\! c{\Delta} t{\sum}_{m = 0}^{P-1}\left( \left\|{\partial\over\partial\tau}(u_1-\tilde u_1)_{m + 1/2}\right\|^{2}_{V_{1}^{\prime}} +\|(u_0-\tilde u_0)_{m + 1/2}\|_V^{2}\right.\\ &&\qquad\qquad \left.+\|(u_1-\tilde u_1)_{m + 1/2}\|_{V_1}^{2}\vphantom{\left\|{\partial\over\partial\tau}\right\|^{2}_{V_{1}^{\prime}}}\right)\\ &&\!+c({\Delta} t)^4+c\|(u_0-\tilde u_0)_{P-1/2}\|_H^{2}+ 2\|z_{0,0}\|_H^{2}+\|(u_0-\tilde u_0)_{1/2}\|_H^{2}\\ &&\!+c{\Delta} t{\sum}_{m = 1}^{P-1}\left\|{(u_0-\tilde u_0)_{m + 1/2}-(u_0-\tilde u_0)_{m-1/2}\over{\Delta} t}\right\|_H^{2}+ \delta T\max_{m = 0,\ldots,M}\|z_{0,M}\|_H^{2}. \end{array} $$

Choosing δ sufficiently small, we have

$$\begin{array}{@{}rcl@{}} \max_{m = 0,\ldots,M}\|z_{0,m}\|_H^{2}&\le& c{\Delta} t{\sum}_{m = 0}^{M-1}\left( \left\|{\partial\over\partial\tau}(u_1-\tilde u_1)_{m + 1/2}\right\|_{V_{1}^{\prime}}^{2} +\|(u_0-\tilde u_0)_{m + 1/2}\|_V^{2}\right.\\ &&\qquad\qquad\left.~ +\|(u_1-\tilde u_1)_{m + 1/2}\|_{V_1}^{2}\vphantom{\left\|{\partial\over\partial\tau}(u_1-\tilde u_1)_{m + 1/2}\right\|_{V_{1}^{\prime}}^{2}}\right)\\ &&+c({\Delta} t)^4+c\max_{m = 1,\ldots,M}\|(u_0-\tilde u_0)_{m-1/2}\|_H^{2}+c\|z_{0,0}\|_H^{2}\\ && +\|(u_0-\tilde u_0)_{1/2}\|_H^{2}\\ &&+c{\Delta} t{\sum}_{m = 1}^{M-1}\left\|{(u_0-\tilde u_0)_{m + 1/2}-(u_0-\tilde u_0)_{m-1/2}\over{\Delta} t}\right\|_H^{2}. \end{array} $$

From this, we get the conclusion.