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\title[Online GMsFEM for Darcy-Forchheimer]{\large Mutiscale model reduction for Darcy-Forchheimer model with online correction in perforated media}


\author[D.A. SPIRIDONOV]{{\bf D.A. Spiridonov}\protect\orcidicon{0000-0002-4081-2703}}



\address{Spiridonov Denis Alekseevich
\newline\hphantom{iii} North-Eastern Federal University,
\newline\hphantom{iii} st. Belinskogo 58,
\newline\hphantom{iii} 677000, Yakutsk, Russia}%
\email{\textcolor{blue}{d.stalnov@mail.ru}}%

\thanks{\sc Spiridonov, D.A.,
Mutiscale model reduction for Darcy-Forchheimer model with online correction in perforated media}
\thanks{\copyright \ 2024 Spiridonov D.A.}
\thanks{\rm The work is supported by the grant of RCF №23-71-10074(https://rscf.ru/project/23-71-10074/)}
\thanks{\it Received Octover, 1, 2024, Published December, ??, 2024}%





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\maketitle {\small

\vspace{-12pt}
\centerline{{\it Communicated by} {\sc P.P. Petrov}
%({\it filled by Editorial board})
}

\bigskip
\bigskip

\begin{quote}

\noindent{\bf Abstract:}In this study, we introduce an algorithm for the Darcy--Forchheimer model in perforated media using the online generalized multiscale finite element method (Online GMsFEM). A nonlinear Darcy flow with a high inertial effect and flow speed is described by the mathematical model. The finite element method (FEM) is used to perform the fine grid approximation. We employ a model reduction technique known as Online GMsFEM, which is predicated on coarse neighborhood local residuals. We employ another offline multiscale basis functions to characterize a boundary condition on perforations in the multiscale approximation. In each local domain, the online multiscale basis functions are built using residuals. The Darcy-Forchheimer equation can now account for the effects of nonlinear coefficients because to the existence of such online multiscale basis functions.  We present  numerical results in a two-dimensional
heterogeneous preforated domain.  We analyze the method's accuracy in relation to the impact of nonlinearity.  The numerical tests demonstrate the excellent accuracy of the Online GMsFEM for these kinds of nonlinear problems. The accuracy weakly depends  on the magnitude of
the nonlinear part of the equation.
\medskip

\noindent{\bf Keywords:} Darcy--Forchheimer model, Online generalized multiscale
finite element method, multiscale model reduction, finite element
method, perforated domain, nonlinear flow
problem.

\noindent{\bf Аннотация:}
В данной работе представлее алгоритм для модели Дарси--Форхгеймера в перфорированных средах с использованием онлайн обобщенного многомасштабного метода конечных элементов (Online GMsFEM). Нелинейный поток Дарси с высоким инерционным эффектом и скоростью потока описывается математической моделью. Метод конечных элементов (МКЭ) используется для аппроксимации мелкой сетки. Применен метод понижения порядка модели, известный как Online GMsFEM, который основан на  невязках в локальный областях грубой сетки. Для учета граничного условия на перфорациях в аппроксимации на грубой сетке строятся специальные оффлайн дополнительные базисные функции. В каждой локальной области онлайн многомасштабные базисные функции строятся с использованием невязок. Уравнение Дарси--Форхгеймера теперь может учитывать эффекты нелинейных коэффициентов благодаря существованию таких онлайн многомасштабных базисных функций. Численные результаты представлены в двумерной неоднородной префорированной области. Точность метода исследуется в зависимости от величины нелинейной части  уравнения. Численные тесты демонстрируют превосходную точность Online GMsFEM для этих видов нелинейных задач. Точность слабо зависит от величины нелинейной части уравнения.

\noindent{\bf Ключевые слова:}
Модель Дарси--Форхгеймера, обобщенный многомасштабный метод конечных элементов,  многомасштабное понижение порядка модели, метод конечных элементов, перфорированная область, нелинейная задача фильтрации.
 \end{quote}
}

\bigskip

\section{Introduction}
In this paper, we consider a multiscale modeling of the
Darcy--Forchheimer model in a perforated medium. In nonlinear
filtration problems, two cases are distinguished. This are high and
low speed flows. At high speed flows, when the inertial component is
significant, the Darcy--Forchheimer formula is used
\cite{girault2008numerical}. Fluid flow in porous media is an
essential element in understanding oil and gas production processes,
as well as in reservoir hydrology, environmental protection, and
many other \cite{cescon2020filtration}.

The basic law of filtration of liquids and gases in  porous media is
Darcy's law; it expresses the dependence of fluid filtration rate on
the pressure gradient. Many scientific works are devoted to checking
and investigating the limits of the applicability of Darcy's law
\cite{gavrilieva2017flow, nikiforov2019numerical}. In cases where the filtration rate is
relatively high, inertial effects cannot be ignored, the
Darcy--Forchheimer model is used \cite{khan2020simulation, chu2021stability}.  High filtration rates can
occur in highly heterogeneous media and near perforations due to infulence of boundary condition.  In this paper, we consider the
multiscale method for the nonlinear flow model in a perforated porous
medium based on the Darcy--Forchheimer model
\cite{girault2008numerical}.


Standard modeling approaches imply the use of  well-known methods of
approximation on a fine grid. One of such methods is well-proven in
the field of mathematical modeling, the finite element method
\cite{dhatt2012finite}. For
modeling the Darcy--Forchheimer model, the mixed finite element
method is most commonly used, providing the law of conservation of
mass \cite{huang2018multigrid, xu2017multipoint}.
In this paper, we use the finite element method to solve a problem
on the fine grid.  

One way is to solve the problem on a coarse grid. One  of these
methods is the homogenization method \cite{stepanov2023prediction, tyrylgin2019numerical}. In this study, we
use multiscale modeling techniques to reduce the dimensionality of
the original problem \cite{efendiev2009multiscale,
allaire2005multiscale}. There are many
modifications of multiscale methods, each modification of the method
may be suitable for a specific task. The most famous is the
multiscale finite element method (MsFEM)
\cite{hou1997multiscale,
masud2006multiscale}. The simple MsFEM has a high error in high
contrast domains. In this regard, a method was invented that allows
adding more basis functions by solving spectral problems, this
method is known as the generalized multiscale finite element method
(GMsFEM) \cite{efendiev2013generalized, spiridonov2019generalized,
spiridonov2020generalized}. Based on the finite volume method, a
multiscale finite volume method (MsFVM) was developed
\cite{vaspir2023, sokolova2019multiscale}. For problems with
complex heterogeneity, such as channels or fractures and perforations, multiscale
methods with an oversampling strategy are well suited, one of such
method is the constraint energy minimizing generalized multiscale
finite element method (CEM-GMsFEM) \cite{chung2018constraint, cheung2020constraint}. Oversampled local
domains allow multiscale basis functions to take into account the
influence of heterogeneity in neighboring local domains. The
Darcy--Forchheimer model is often written in a mixed formulation, a
mixed  finite element method (Mixed-FEM) has been developed to solve
such problems \cite{spiridonov2022mixed}. Another approach of multiscale modeling of
Darcy--Focrhheimer flow is Non-local multi-continuum method (NLMC)
\cite{spiridonov2024non}. For nonlinear problems, the online
generalized multiscale finite method (Online GMsFEM) is well suited
that allows to take into account changes in properties by adding
additional multiscale bases while solving the problem
\cite{spiridonov2023online, spiridonov2021online,
chung2015residual}.

This paper considers an Online generalized multiscale finite element
method (Online GMsFEM) for Darcy--Forchheimer flow in a
heterogeneous perfrated domain. The algorithm consists of two parts:
offline and online stages. In the offline stage, we construct an
offline multiscale basis functions, which do not change throughout
the entire modeling process. Also, in the offlien stage we calculate an additional multiscale basisd functions to take into account boundary conditions on perforations.  We solve the system on a coarse grid
and construct a residual-based online multiscale basis functions at
the online stage using the obtained bases. Note that the
approximation is based on the finite element method.
Numerical results are presented for a two-dimensional perforated
heterogeneous domain. The numerical experiment includes
investigating the nonlinearity effects on the Online GMsFEM
accuracy. We note that one can, in general, design various online
approaches for nonlinear problems. Our goal is to have a simple and
efficient online approach, which was proposed here.

The paper is organized as follows. Section 2 presents the
Darcy--Forchheimer model in a perforated heterogeneous domain.
Section 3 describes a  fine grid approximation using the finite
element method. Section 4
presents an algorithm of the online generalized multiscale finite
element method (Online GMsFEM) for the Darcy--Forchheimer model. We
provide a numerical experiment to investigate the influence of
nonlinearity on the method accuracy in Section 5.


\section{Problem formulation}
The filtration process is described by the well-known Darcy law
equation
\begin{equation} \label{eq1}
\mu k^{-1} u+\nabla p = 0. \nonumber
\end{equation}
Let us supplement the basic equation of Darcy's law with a nonlinear
term. And we can obtain a Darcy--Forchheimer single phase flow model
\begin{equation} \label{eq2}
\begin{split}
\mu k^{-1} u + \rho \beta |u|u + \nabla p = g, \quad x \in \Omega, \\
\nabla \cdot u = f, x \in \Omega,
\end{split}
\end{equation}
where $p$ is the pressure, $u$ is the velocity, $k$ is the
heterogeneous permeability,  $\mu$ is the viscosity, $\rho$ is the
density and $\beta$ is the Forchheimer coefficient. The impact of
nonlinearity on the overall physical process is determined by the
Forchheimer coefficient.

The system \eqref{eq2} is complemented with Dirchlet boundary condition,
\begin{equation} \label{eq3}
p = p_D, \quad x \in \partial \Omega. \nonumber
\end{equation}

In the continuous level, the Darcy--Forchheimer equation can be
rewritten into a nonlinear primary formulation. For simplicity, we
assume that the permeability is a scalar. Taking the norm of the
Darcy--Forchheimer equation, we obtain
\begin{equation}
\begin{split}
(\mu k^{-1}+\rho \beta |u|)|u|=|-\nabla p + g|, \\
\rho \beta |u|^2 + \mu k^{-1} |u|-|\nabla p - g| = 0,
\end{split} \nonumber
\end{equation}
and can solve for $|u|$
\begin{equation}
|u| = \frac{-\mu k^{-1} + \sqrt{(\mu k^{-1})^2 + 4 \rho \beta |\nabla p - g|}}{2\rho \beta}, \nonumber
\end{equation}
and consequently $u$:
\begin{equation}
u=-\frac{\nabla p - g}{\mu k^{-1}+\rho \beta |u|}=-\frac{2(\nabla p - g)}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta|\nabla p - g|}}. \nonumber
\end{equation}
Then, substituting back to the divergence equation, we get the
primary formulation of pressure $p$ only
\begin{equation}\label{eq4}
-\nabla \cdot \Bigg(\frac{2(\nabla p - g)}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta|\nabla p - g|}}\Bigg)=f.
\end{equation}
The well-posedness of primary formulation of pressure can be found in \cite{pan2012mixed}.

In this study, we take a dynamic nonlinear  filtration process into
consideration and include a temporal derivative and $g=0$.
Therefore, the dynamic incompressible single-phase
Darcy--Forchheimer flow equation can be written as follows
\begin{equation}\label{eq5}
c\frac{\partial p}{\partial t} -\nabla \cdot \Bigg(\frac{2\nabla p}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta|\nabla p|}}\Bigg)=f.
\end{equation}

In this paper, we consider the nonlinear filtration process in
fractured media. We define the porous matrix domain as $\Omega \in
\mathcal{R}^d$. In our implementation, we consider problems in
the two-dimensional domain $d=2$.
And we can define the nonlinear Darcy--Forchheimer flow  in a perforated
domain:
\begin{equation} \label{eq6}
c \frac{\partial p}{\partial t} + \nabla \cdot  u = f, \quad x \in \Omega, \quad t > 0,
\end{equation}
where,  
\begin{equation} \label{eq7}
u = - \frac{2\nabla p}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta|\nabla p|}}. \nonumber
\end{equation}
We apply the following vounary condition for \eqref{eq6}:
\begin{equation} \label{eq8}
p = g, \quad x \in \Gamma_P, \quad p = 0, \quad x \in \partial \Omega/\Gamma_P,
\end{equation}
and set the initial condition $p = p_0$ for $t = 0$, $\Gamma _P$ is the  boundary of  perforations inside domain $\Omega$.


\section{Fine grid approximation}


Next, we consider the fine grid approximation  of the problem
\eqref{eq6}, \eqref{eq8}. We use the finite element method with a
discrete fracture model. We define unstructured triangular fine grid
$\mathcal{T}_h$, with perforations. For
the time-dependent problem,  we note $n$ as a number of time layers,
$\tau$ as the time step, and $T_{max}=n\tau$ as the final time. We
consider $N_f$ as the number of elements in fine grid.

We have the following approximation on the fine grid:
\begin{equation}\label{eq10}
\begin{split}
\int_{\Omega}  c \frac{\partial p}{\partial t}\, v     \, dx +\int_{\Omega}
\left( \frac{2\nabla p}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta|\nabla p|}}, \nabla v \right) dx= \\
 =\int_{
\Omega} f \, v dx, \quad v \in \hat{V}, 
\end{split}
\end{equation}
where $p  \in V$ and
\[
V = \{v \in H^1(\Omega ): v = g , \, x \in \Gamma_P , \, v = 0 , \, x \in \partial\Omega / \Gamma_P  \}, \]
\[
\hat{V} = \{v \in H^1(\Omega): v = 0 , \, x \in \partial \Omega \}.
\]
For the approximation of the time derivative, we use the backward
Euler method. To~resolve non-linearity  we use Picard iteration
method and obtain the following approximation on the fine grid
\begin{equation}\label{eq11}
\begin{split}
\int_{\Omega}  c \frac{p^{n+1}_{t+1}-p^n}{\tau}\, v  \, dx +\int_{\Omega}  \left( \frac{2\nabla p^{n+1}_{t+1}}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta|\nabla p^{n+1}_{t}|}}, \nabla v \right) dx = \\
=\int_{
\Omega} f \, v dx, \quad v \in \hat{V}, \\
\end{split}
\end{equation}
where $t$ denotes a Picard iteration step. Note that $p^n$ is no subindex $t$ since they show a previous time iterate.
We use the following termination criteria in the Picard iterations:
\begin{equation}\label{eq12}
\frac{||p^{n+1}_{t+1}-p^{n+1}_{t}||_{L_2}}{||p^{n+1}_{t+1}||_{L_2}} < \varepsilon , \nonumber
\end{equation}
in our implementation, we took $\varepsilon = 10^{-3}$.

We can write approximation, in~the matrix form as
\begin{equation}
\label{matrix}
\frac{1}{\tau} S_{t+1}^{n+1}(p_{t+1}^{n+1} - p^n) + A_{t}^{n+1} p_{t+1}^{n+1} = F^{n+1}_{t+1}
\end{equation}
where $S^{n+1}_{t+1}$ is a mass matrix that  corresponds to the
first term, $A^{n+1}_{t}$ is a stiffness matrix that corresponds
to the second nonlinear term and $F^{n+1}_{t+1}$ is
a source term in \eqref{eq11}.


\section{Coarse grid approximation}

To make approximation on the coarse grid, we apply an Online
Generalized Multiscale Finite Element method.  A coarse grid
$\mathcal{T}_H$ is denoted by
\begin{equation}
\mathcal{T}_H = \bigcup_j K_j, \nonumber
\end{equation}
here $K_j$ is the cell of coarse grid. Following that, we create a
local coarse neighborhood domain $\omega _i$ it is produced by
merging all coarse cells around one coarse grid vertex:
\begin{equation}
\omega _i=\bigcup_j{K_j \in \mathcal{T} \ : \ x_i \in K_j}, \nonumber
\end{equation}
where $i=\overline{1,N_c}$ and $N_c$ -- the number of coarse grid vertices.

We can identify two stages in GMsFEM, online and offline.


\subsection{Offline~Stage}
We start from the constructing a snapshot space $V^{\omega
_i}_{snap}$.  The~snapshot space are obtained by solving next local
problems
\begin{equation} \label{e7}
\int_{\omega _i} \Bigg(\frac{2\nabla \phi^i_{j}}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta}}, \nabla w^i\Bigg) dx = 0, \nonumber
\end{equation}
with boundary condition:
\begin{equation}
\phi^i_{j} = \delta_j, \quad x \in \partial \omega_i / \Gamma_P, \quad \frac{\partial \phi^i}{\partial n} = 0, \quad  x \in \Gamma_P,
\end{equation}
where  $\delta_j$ is the discrete delta function which takes the value $1$ at the $j$-th fine grid node $x=x_j$ and zero elsewhere ($j=1,...,J^i$, $J^i$ is number of fine grid nodes on boundary $\partial \omega _i$ ).


The snapshot space and the projection matrix on the snapshot space are defined as follows
\[
V^{snap}_i = \text{span}\{ \phi^{i}_{1}, ...,\phi^{i}_{J_i}\}, \quad \text{ and } \quad
R^{snap}_{i} = ( \phi^{i}_1, ...,\phi^{i}_{J_i} )^T.
\]

On the snapshot space we solve the next local spectral problem in
each $\omega _i$ to  obtain an offline multiscale basis functions
\begin{equation} \label{e4}
\tilde{A} \tilde{\Psi}^i_{snap,j} = \lambda \tilde{S} \tilde{\Psi}^i_{snap,j}, \nonumber
\end{equation}
with
\[
\tilde{A}  = R_{snap,i} A^{\omega_i} R^T_{snap,i}, \quad
\tilde{S}  = R_{snap,i} S^{\omega_i} R^T_{snap,i},
\]
where
\begin{equation}
\label{e5}
\begin{split}
A^{\omega_i}= \{ a_{ln} \}, \quad
a_{ln} = \int_{\omega _i} \Bigg(\frac{2\nabla \psi_{l}}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta}}, \nabla \psi_{n}\Bigg) dx , \\
S^{\omega_i}= \{ s_{ln} \}, \quad
s_{ln}  = \int_{\omega _i}\frac{2\nabla \psi_{l}}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta}} \psi_{n} \, dx.
\end{split} \nonumber
\end{equation}

We use solution of the spectral problem $\Psi^i_j =
R_{snap,i}\tilde{\Psi}^i_{snap,j}$ in  offline basis construction.
We should note that, we construct the offline multiscale basis
functions only once in the offline stage. Because of that, we take
$|\nabla p^{n+1}_{t}|=1$ for offline bases computation. We obtain offline multiscale basis
functions by multiplication on the linear partition of unity
function $ \psi^i_j = \chi_i \Psi^i_j$, {where $\chi_i$ is the
linear coarse grid nodal basis function that is equal to zero on the
boundary of local domain $\omega _i$ and one at the  coarse grid
node $i$.}

Now, we can determine the offline space
\[
V_{ms} = \mbox{span}(
\psi^1_1, ... ,\psi^1_{M_1}, ....,
\psi^{N_c}_1, ...,\psi^{N_c}_{M_{N_c}}).
\]
and the offline projection matrix
\[
R^T = (
\psi^1_1, ... ,\psi^1_{M_1}, ....,
\psi^{N_c}_1, ...,\psi^{N_c}_{M_{N_c}}).
\]

\textbf{Additional offline multiscale basis functions.}

Inside the computation domain $\Omega$ we have parforations. And to take into account the influence of boundary condition $\Gamma _P$ we need to introduce an additional basis function in the offline stage. This function we denote as $\theta^i$ and we calculte it only in local domains $\omega _i$ which contains perforations. Let $N_P$ is the number of $\omega_i$ with perforations. To obtain an additional basis function we solve the next problem in $\omega _i$ with perforations:
\begin{equation}\label{addbass}
\begin{split}
 -\nabla \cdot \Bigg(\frac{2\nabla \theta ^i}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta}}\Bigg) &=0, \quad x\in \omega_i, \\
 \theta ^i = 0, \quad x \in \partial \omega _i / \Gamma _p, \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ &  \\ 
  \theta ^i = g, \quad x \in  \Gamma _p.  \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \  \ \ \ \  \ \   &
\end{split}
\end{equation} 

After that, we include additional basis functions in multiscale space $V_{ms}$ and projection matrix $R$:
\[
V_{ms} = \mbox{span}(
\psi^1_1, ... ,\psi^1_{M_1}, ....,
\psi^{N_c}_1, ...,\psi^{N_c}_{M_{N_c}}, \theta^1, ..., \theta^{N_P}),
\]\[
R^T = (
\psi^1_1, ... ,\psi^1_{M_1}, ....,
\psi^{N_c}_1, ...,\psi^{N_c}_{M_{N_c}}, \theta^1, ..., \theta^{N_P}).
\]

\subsection{Online~Stage}

{In online GMsFEM,  we update projection matrix $R^n$ by  adding
online residual based multiscale basis functions at $n$-th time
step.   Therefore,  we solve next system on the coarse~grid}
\begin{equation}
\label{eq:mfa-on}
S_{c,t+1}^{n+1} \frac{ p_{c,t+1}^{n+1} -  p_c^{n} }{\tau} +
A_{c,t}^{n+1} p_{c,t+1}^{n+1} = F_{c,t+1}^{n+1},
\end{equation}
where coarse scale matrices and vectors are constructed using current projection matrix $R^{n+1}$
\[
S_{c,t+1}^{n+1} = R^{n+1} S_{t+1}^{n+1}  (R^{n+1})^T, \quad
A_{c,t}^{n+1} = R^{n+1} A_t^{n+1}  (R^n)^T,\]\[
F_{c,t+1}^{n+1} = R^{n+1}  F_{t+1}^{n+1}, \quad
p^{n+1}_{ms} = (R^{n+1})^T p^{n+1}_c.
\]

Note that, when we use pre-constructed offline multiscale basis
functions for given heterogeneity  and fracture distribution in
coarse scale system construction,  we can use predefined  projection
matrix without updating it at $n$-th time step
\[
(R^n)^T =
R^T = (
\psi^1_1, ... ,\psi^1_{M_1}, ....,
\psi^{N_c}_1, ...,\psi^{N_c}_{M_{N_c}}, \theta^1, ..., \theta^{N_P}).
\]

In online GMsFEM, we set $R^n = R$ at first time step ($n = 0$),
then we construct and solve coarse scale system \eqref{eq:mfa-on}.
Next,  at~time step $n = 1,2,...$, we enrich multiscale space by
residual based online multiscale basis functions.  In~order to
enrich space, we solve system \eqref{eq:mfa-on} with $R^n =
R^{n-1}$,  then we calculate online multiscale basis functions
$\vartheta^i_1$ locally  in $\omega_i$ using current residuals and
update the projection matrix
\[
(R^{1,n})^T = (
\psi^1_1, ... ,\psi^1_{M_1}, ....,
\psi^{N_c}_1, ...,\psi^{N_c}_{M_{N_c}},
\vartheta_1^1, ..., \vartheta_1^{N_c}
).
\]

We can reiterate the process with residual  calculation and add more
online basis functions on the $n$th time step
\[
(R^n)^T = (R^{L,n})^T = (
\psi^1_1, ... ,\psi^1_{M_1}, ....,
\psi^{N_c}_1, ...,\psi^{N_c}_{M_{N_c}}, \theta^1, ..., \theta^{N_P},\] \[
\vartheta_1^1, ..., \vartheta_1^{N_c}
...,
\vartheta_L^1, ..., \vartheta_L^{N_c}
),
\]
where $L$ is the number of online iterations.  We can enrich
multiscale  space not for every time steps, for~example, every 4-th.
Then, when we keep $R^n$ for the time steps where we do not want to
update online basis functions. Next, we present construction of the
online multiscale basis functions in local domain $\omega_i$
in~details.

Construction of the local residual based  online multiscale basis
functions is based on the solution of the following local problem in
$\omega_i$:
\begin{equation}
\label{online}
\begin{split}
a_{\omega_i}(\vartheta^i_l, v) = r^l_{\omega_i}(v),  \quad l = 1,...,L,
\end{split} \nonumber
\end{equation}
where
\begin{equation}
\label{online2}
\begin{split}
a_{\omega_i}(\vartheta^i_l, v) &=
\int_{\omega _i} c  \frac{\vartheta^i_{l}}{\tau}   \, v   \, dx
+ \int_{\omega _i}  \left( \frac{2\nabla \vartheta^i_{l}}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta|\nabla p^{l,n+1}_{ms,t+1}|}}, \nabla v \right)  dx + \\
r^l_{\omega_i}(v)
&= \int_{\omega _i} f \, v \, dx - \int_{\omega _i} c  \frac{p_{ms,t+1}^{l, n+1} - p_{ms}^{l,n}}{\tau}   \, v    \, dx  -
\\&- \int_{\omega _i}  \left( \frac{2\nabla p_{ms,t+1}^{l, n+1}}{\mu k^{-1}+\sqrt{(\mu k^{-1})^2+4\rho \beta|\nabla p^{l,n+1}_{ms,t+1}|}} , \nabla v \right) dx 
\end{split} \nonumber
\end{equation}
with the following Dirichlet boundary conditions:
\begin{equation} \label{BoundDir}
\vartheta_{l}^{i} = 0, \quad x \in \partial \omega _i / \Gamma_P, \quad  \vartheta_{l}^{i} = g, \quad x \in \Gamma_P.
\end{equation} 

Using constructed online multiscale basis functions,  we enrich the
multiscale space by adding online basis functions {$\vartheta_l^i$}:
 \[
V_{ms} = \mbox{span}(
\psi^i_1, ... ,\psi^i_{M_i}, \theta^1, ..., \theta^{N_P},
\vartheta_l^1, ..., \vartheta_l^{N_c},
\,  i = 1,...,N_c,  \,  l =1,...,L.).
\]

Here we can add several online multiscale basis functions in local
domain $\omega_i$ based on the current solution $p^{l,n}_{ms}$
\begin{equation}
\label{eq:mfa}
S_{c,t+1}^{l, n+1} \frac{ p_{c,t+1}^{l,n+1} -  p_c^n }{\tau} +
A_{c,t}^{l,n} p_{c,t+1}^{l, n+1} = F_{c,t+1}^{l,n+1}, \nonumber
\end{equation}
{where coarse scale matrices and vectors are constructed using current projection matrix $R^{l, n}$}
\[
S_{c,t+1}^{l,n+1} = R^{l,n+1} S^{n}_{t+1}  (R^{l,n+1})^T, \quad
A_{c,t}^{l,n+1} = R^{l,n+1} A^{n+1}_t  (R^{l,n+1})^T, \quad \]
\[
F_{c,t+1}^{l,n+1} = R^{l,n+1}  F^{n+1}_{t+1}, \quad
p^{l,n+1}_{ms,t+1} = (R^{l,n+1})^T p^{l, n+1}_{c,t+1}.
\]
with projection matrix
\[
R^{l, n+1} = (
\psi^1_1, ... ,\psi^1_{M_1}, ....,
\psi^{N_c}_1, ...,\psi^{N_c}_{M_{N_c}}, \theta^1, ..., \theta^{N_P},
\vartheta_1^1, ..., \vartheta_1^{N_c}....,
\vartheta_l^1, ..., \vartheta_l^{N_c}
)^T,
\]
where $l = 1,...,L$,  $R^{n+1} = R^{L, n+1},$ and $L$ is the number
of the online~iterations.


\section{Numerical results}

In this section, we use Online GMsFEM to provide numerical results for the Darcy--Forchheimer problem.
We consider the flow problem in the two-dimensional perforated heterogeneous domain.
We show the heterogeneous coefficient $k_m$ in Figure \ref{domkle}.  %Because we evaluate the worst-case scenarios in these test cases, we can demonstrate the method's applicability.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9\linewidth]{figs/kle.png}
\end{center}
\caption{Heterogeneity and computational domain $\Omega$. Left: computational domain $\Omega$. Right: Coefficient $k_m$.}
\label{domkle}
\end{figure}

We consider numerical experiments for coarse grid $10 \times 10$ and
fine  grid with $14649$ vertices. We consider results with different
values of $\beta$ to investigate the effect of nonlinearity. In
the implementation, we consider results for $\beta=10^5$, $\beta=10^4$,
$\beta=10^3$, $\beta=10^2$, $\beta=10$. And we take
maximum time $T_{max}=1.5, \ T_{max}=0.5, \ T_{max}=0.17, \ T_{max}=0.05, \
T_{max}=0.017$ respectively for each value of $\beta$ and we use 20
time layers for each test case. We set $\mu=1.0$, $\rho=1.0$,
$c=1.0$, $f=0$. We put Dirichlet boundary
condition on perforations $p=g, g=1.0$ and zero Dirichlet boundary on the all external boundaries of $\Omega$.  To investigate the accuracy of the Online GMsFEM algorithm, we compare
the multiscale solutions with the reference solutions. We take a
fine-grid solution as a reference solution. The fine grid system's
size equals $14649 \times 14649$.

We compare solutions by using $L_2$ and $H_1$ norm errors
\begin{equation} \label{eq17}
e^{L_2}=\sqrt{\frac{\int_{\Omega}(p_f-p_{ms})^2
dx}{\int_{\Omega}p^2_f dx}} \cdot 100 \%,   \quad
e^{H_1}=\sqrt{\frac{\int_{\Omega}(\nabla(p_f-p_{ms}))^2
dx}{\int_{\Omega}(\nabla p)^2_f dx}} \cdot 100 \%, \nonumber
\end{equation}
where $p_{ms}$ is the multiscale solution, $p_f$ is the solution on the fine grid.



\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9\linewidth]{figs/results.png}
\end{center}
\caption{Numerical results for $\beta = 10^5$ at the 2-nd, 10-th, and last time layer. First row: fine grid solution. Second row: multiscale solution on using only 12 offline multiscale basis functions. Third row: multiscale solution on using 12 offline multiscale basis functions and 2 online multiscale basis function.}
\label{results1}
\end{figure}


The numerical results are presented in Fig.~\ref{results1} for
$\beta=10^5$. The first row shows a solution on a fine grid, the
second row shows a solution using GMsFEM using only 12 offline
multiscale basis functions without online correction, the third row
shows a solution using Online GMsFEM using 12 offline multiscale
basis functions and 2 online multiscale basis function. We present
solutions for the 2-nd, 10-th, and final time layers to demonstrate
time distribution.  In the figures, we can see small oscillations in
the offline GMsFEM solution near perforations. Online GMsFEM
solution and fine grid solution looks very similar, in the Online
GMsFEM solution we don't see any oscillations.


\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
 Number of & \multicolumn{3}{|c|}{Offline basis} & \multicolumn{3}{|c|}{1 Online basis} & \multicolumn{3}{|c|}{2 Online basis} \\
\cline{2-10}
offline basis  
& \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} & \multirow{2}{*}{$DOF_c$ }  & \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} &  \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} \\
functions & & & & & & & & & \\
\hline
1 & 202 & 5.951 & 76.667 & 323 & 6.289 & 48.488 & 444 & 3.145 & 33.133 \\
2 & 323 & 5.281 & 73.391 & 444 & 5.525 & 45.693 & 565 & 2.777 & 31.578 \\
4 & 569 & 3.054 & 54.298 & 686 & 2.847 & 29.362 & 806 & 1.872 & 22.797 \\
6 & 807 & 2.731 & 52.331 & 928 & 2.241 & 24.967 & 1049 & 1.538 & 19.927 \\
8 & 1049 & 2.249 & 50.524 & 1170 & 1.717 & 21.179 & 1291 & 1.271 & 17.653 \\
12 & 1533 & 2.026 & 49.641 & 1654 & 1.357 & 18.943 & 1775 & 0.962 & 15.971 \\
16 & 2017 & 1.905 & 49.065 & 2138 & 1.093 & 17.201 & 2259 & 0.741 & 14.418 \\
24 & 2985 & 1.788 & 48.412 & 3106 & 0.807 & 14.874 & 3227 & 0.568 & 12.541 \\
32 & 3953 & 1.756 & 48.348 & 4074 & 0.743 & 14.481 & 4195 & 0.492 & 12.014 \\
\hline
\end{tabular}
\end{center}
\caption{Numerical results for $\beta = 10$. Relative $L_2$ and $H_1$($\%$) errors with different number of offline multiscale basis functions using 1 and 2 online basis functions on the last time layer.}
\label{table1}
\end{table}

\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
 Number of & \multicolumn{3}{|c|}{Offline basis} & \multicolumn{3}{|c|}{1 Online basis} & \multicolumn{3}{|c|}{2 Online basis} \\
\cline{2-10}
offline basis  
& \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} & \multirow{2}{*}{$DOF_c$ }  & \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} &  \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} \\
functions & & & & & & & & & \\
\hline
1 & 202 & 6.261 & 76.274 & 323 & 5.879 & 48.605 & 444 & 3.481 & 31.172 \\
2 & 323 & 5.251 & 73.037 & 444 & 5.561 & 45.345 & 565 & 3.165 & 29.598 \\
4 & 569 & 3.007 & 53.975 & 686 & 2.875 & 29.548 & 806 & 1.874 & 22.111 \\
6 & 807 & 2.688 & 52.035 & 928 & 2.254 & 24.811 & 1049 & 1.529 & 19.268 \\
8 & 1049 & 2.234 & 50.272 & 1170 & 1.731 & 21.063 & 1291 & 1.251 & 17.106 \\
12 & 1533 & 2.024 & 49.389 & 1654 & 1.361 & 18.796 & 1775 & 0.967 & 15.435 \\
16 & 2017 & 1.917 & 48.821 & 2138 & 1.098 & 17.029 & 2259 & 0.767 & 13.946 \\
24 & 2985 & 1.801 & 48.179 & 3106 & 0.808 & 14.741 & 3227 & 0.565 & 12.299 \\
32 & 3953 & 1.777 & 48.124 & 4074 & 0.747 & 14.361 & 4195 & 0.497 & 11.745 \\
\hline
\end{tabular}
\end{center}
\caption{Numerical results for $\beta = 10^2$. Relative $L_2$ and $H_1$($\%$) errors with different number of offline multiscale basis functions using 1 and 2 online basis functions on the last time layer.}
\label{table2}
\end{table}

\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
 Number of & \multicolumn{3}{|c|}{Offline basis} & \multicolumn{3}{|c|}{1 Online basis} & \multicolumn{3}{|c|}{2 Online basis} \\
\cline{2-10}
offline basis  
& \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} & \multirow{2}{*}{$DOF_c$ }  & \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} &  \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} \\
functions & & & & & & & & & \\
\hline
1 & 202 & 5.991 & 76.613 & 323 & 6.294 & 48.657 & 444 & 3.933 & 32.897 \\
2 & 323 & 5.355 & 73.374 & 444 & 5.592 & 45.477 & 565 & 3.619 & 31.598 \\
4 & 569 & 3.077 & 54.225 & 686 & 2.868 & 29.411 & 806 & 2.029 & 23.141 \\
6 & 807 & 2.731 & 52.258 & 928 & 2.255 & 24.944 & 1049 & 1.631 & 19.862 \\
8 & 1049 & 2.252 & 50.471 & 1170 & 1.721 & 21.161 & 1291 & 1.307 & 17.552 \\
12 & 1533 & 2.032 & 49.568 & 1654 & 1.357 & 18.925 & 1775 & 1.011 & 15.736 \\
16 & 2017 & 1.918 & 49.004 & 2138 & 1.102 & 17.212 & 2259 & 0.811 & 14.172 \\
24 & 2985 & 1.789 & 48.341 & 3106 & 0.813 & 14.865 & 3227 & 0.593 & 12.501 \\
32 & 3953 & 1.761 & 48.277 & 4074 & 0.746 & 14.463 & 4195 & 0.528 & 11.934 \\
\hline
\end{tabular}
\end{center}
\caption{Numerical results for $\beta = 10^3$. Relative $L_2$ and $H_1$($\%$) errors with different number of offline multiscale basis functions using 1 and 2 online basis functions on the last time layer.}
\label{table3}
\end{table}



\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
 Number of & \multicolumn{3}{|c|}{Offline basis} & \multicolumn{3}{|c|}{1 Online basis} & \multicolumn{3}{|c|}{2 Online basis} \\
\cline{2-10}
offline basis  
& \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} & \multirow{2}{*}{$DOF_c$ }  & \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} &  \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} \\
functions & & & & & & & & & \\
\hline
1 & 202 & 5.911 & 76.307 & 323 & 6.256 & 48.572 & 444 & 4.312 & 35.833 \\
2 & 323 & 5.294 & 73.099 & 444 & 5.586 & 45.344 & 565 & 4.032 & 34.681 \\
4 & 569 & 3.028 & 54.005 & 686 & 2.892 & 29.594 & 806 & 2.182 & 24.134 \\
6 & 807 & 2.701 & 52.059 & 928 & 2.271 & 24.858 & 1049 & 1.757 & 20.488 \\
8 & 1049 & 2.243 & 50.305 & 1170 & 1.737 & 21.094 & 1291 & 1.378 & 17.875 \\
12 & 1533 & 2.031 & 49.394 & 1654 & 1.366 & 18.814 & 1775 & 1.061 & 15.862 \\
16 & 2017 & 1.929 & 48.845 & 2138 & 1.114 & 17.111 & 2259 & 0.848 & 14.209 \\
24 & 2985 & 1.804 & 48.184 & 3106 & 0.815 & 14.763 & 3227 & 0.617 & 12.504 \\
32 & 3953 & 1.778 & 48.118 & 4074 & 0.751 & 14.371 & 4195 & 0.543 & 11.936 \\
\hline
\end{tabular}
\end{center}
\caption{Numerical results for $\beta = 10^4$. Relative $L_2$ and $H_1$($\%$) errors with different number of offline multiscale basis functions using 1 and 2 online basis functions on the last time layer.}
\label{table4}
\end{table}


\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
 Number of & \multicolumn{3}{|c|}{Offline basis} & \multicolumn{3}{|c|}{1 Online basis} & \multicolumn{3}{|c|}{2 Online basis} \\
\cline{2-10}
offline basis  
& \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} & \multirow{2}{*}{$DOF_c$ }  & \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} &  \multirow{2}{*}{$DOF_c$ }  
& \multirow{2}{*}{$e^{L_2}$} & \multirow{2}{*}{$e^{H_1}$} \\
functions & & & & & & & & & \\
\hline
1 & 202 & 5.888 & 76.288 & 323 & 6.102 & 48.208 & 444 & 4.517 & 39.052 \\
2 & 323 & 5.273 & 73.083 & 444 & 5.455 & 45.082 & 565 & 4.224 & 37.937 \\
4 & 569 & 2.996 & 53.986 & 686 & 2.882 & 29.519 & 806 & 2.288 & 25.371 \\
6 & 807 & 2.671 & 52.044 & 928 & 2.312 & 25.356 & 1049 & 1.858 & 21.651 \\
8 & 1049 & 2.212 & 50.291 & 1170 & 1.763 & 21.559 & 1291 & 1.439 & 18.765 \\
12 & 1533 & 1.996 & 49.376 & 1654 & 1.393 & 19.404 & 1775 & 1.117 & 16.879 \\
16 & 2017 & 1.901 & 48.838 & 2138 & 1.156 & 17.913 & 2259 & 0.911 & 15.478 \\
24 & 2985 & 1.771 & 48.168 & 3106 & 0.871 & 15.695 & 3227 & 0.713 & 13.892 \\
32 & 3953 & 1.742 & 48.086 & 4074 & 0.813 & 15.339 & 4195 & 0.653 &  13.477 \\
\hline
\end{tabular}
\end{center}
\caption{Numerical results for $\beta = 10^5$. Relative $L_2$ and $H_1$($\%$) errors with different number of offline multiscale basis functions using 1 and 2 online basis functions on the last time layer.}
\label{table5}
\end{table}


We present relative $L_2$ and $H_1$ errors in Tables \ref{table1}-\ref{table5} for multiscale solution using 0, 1, and 2 online basis functions for different number of offline basis functions. In the tables, $DOF_c$ denotes the vector size on the coarse grid. By this tables we can see that, the accuracy behavior is similar for all values of $\beta$. In all test cases, when we use 12 offline and 1 online basis functions we can obtain a solution with accuracy good accuracy. But, without online correction, we can't achieve high accuracy, we have very large value of $H_1$ error. 
We can conclude that it is necessary to use online correction in this problem. Using online basis functions, we can obtain a solution using a system with a smaller number of unknowns. Adding a second online basis in each $\omega _i$ improves the accuracy of the method very slightly. Overall, we can improve the accuracy of the method by about 4 times.

We need to discuss the results of the numerical experiment.  We were able to prove that the proposed algorithm of the Online GMsFEM has good accuracy in all performed calculations.  The method showed good accuracy in a highly heterogeneous perforated domain. The accuracy of the method is practically independent of the influence of the nonlinear part of the equation. An online correction procedure greatly improve the accuracy of the method. The method obtains the exact solution on a coarser grid. Moreover, the dimension of the original system differs significantly. From here, we can notice significant savings in computational resources, which is the main advantage of multiscale methods over standard mathematical modeling approaches.  



\section{Conclusion}

This paper presents an Online generalized multiscale finite element method (Online GMsFEM) algorithm for the time-dependent Darcy-Forchheimer model in a perforated heterogeneous domain.  We construct the fine grid approximation  using the finite element method. Model and method formulations were given for two-dimensional cases.  We investigated nonlinearity's impact by varying the coefficient $\beta$ in the numerical experiments. The numerical experiment showed that the proposed approach  provided accurate results without significant influence of the nonlinear part of the flow.   We considered a multiscale method with different numbers of offline and online basis functions. Our experiments showed that online-correction procedure significantly reduces the error of the method, especially in $H_1$ norm. Adding 2-nd online basis functions  gives very little improvement in the accuracy, from here we conclude that, it is not necessary to use the 2-nd online basis.  
The performed numerical experiment showed good accuracy of the method. We conclude that the porosed algorithm of Online generalized multiscale finite element method performed well in modeling the Darcy-Forchheimer model in the perforated medium.



%\section{Acknowledgements}

%D.S. Spiridonov's work is supported is supported by the grant of Russian Science Foundation No. 23-71-30013(https://rscf.ru/project/23-71-30013/) and within the framework of Grant No. FSRG-2024-0003 of the Ministry of Science and Higher Education
%of the Russian Federation. 



\bigskip
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\end{thebibliography}
\end{document}
