In this section, we present some theoretical considerations in space-time coupled and space-time decoupled finite element methods that are necessary to design a possible a posteriori error computation strategy in these methods for initial value problems. We present details in the following.

Let

$A\varphi -f=0\text{in}{\Omega }_{xt}={\Omega }_x\times {\Omega }_t$(1)

be an initial value problem over the space-time domain ${\Omega }_{xt}$ in which $A$ is the space-time differential operator containing $2m_1$ and $2m_2$ as the highest orders of derivatives in space and time. Let ${\bar{\Omega }}_{xt}={\Omega }_{xt}\cup \Gamma$, in which $\text{Γ}$ is the closed boundary of ${\Omega }_{xt}$. Surana et al. [1] have shown that space-time differential operators are non self-adjoint or non-linear but not self-adjoint as in space-time differential equations, $A^{\text{*}}$ adjoint of the operator is not the same as the operator $A$ and there are issues in the concomitant due to open boundary. Thus, for initial value problems, all space-time integral form based on fundamental lemma (Space-time Galerkin Method, Space-time Galerkin Method with Weak Form, Space-time Petrov-Galerkin Method, Space-time Weighted Residual Method) are space-time variationally inconsistent due to lack of unique extremum principle. Thus, for initial value problems, only the integral form based on space-time residual functional are space-time variationally consistent [1] , hence are considered in the present work for a posteriori error computations.

Let ${\left({\bar{\Omega }}_{xt}^{\left(n\right)}\right)}^T={\displaystyle \underset{e}{\cup }{\bar{\Omega }}_{xt}^{\left(e\right)}}$ be discretization of the $n^{th}$ space-time strip or the $n^{th}$ space-time slab ${\bar{\Omega }}_{xt}^{\left(n\right)}$ in which ${\bar{\Omega }}_{xt}^{\left(e\right)}$ is a $p$-version hierarchical space-time finite element (see Figure 1 ). Let ${\varphi }_h^e\left(x,t\right)$ be local approximation of $\varphi$ over ${\bar{\Omega }}_{xt}^{\left(e\right)}$ given by

${\varphi }_h^e\left(x,t\right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{N}_i\left(x,t\right){\delta }_i^e$(2)

in which $N_i\left(x,t\right)$ are space-time local approximation functions for an element ${\bar{\Omega }}_{xt}^{\left(e\right)}$ derived using its map in the natural coordinate space $\xi$, $\eta$ and $\zeta$ in a one unit length, one unit square or a one unit cube in ${ℝ}^1$, ${ℝ}^2$ and ${ℝ}^3$ respectively.

Let

${\varphi }_h\left(x,t\right)={\displaystyle \underset{e}{\cup }{\varphi }_h^e\left(x,t\right)}$(3)

be approximation of $\varphi \left(x,t\right)$ over the discretization ${\left({\bar{\Omega }}_{xt}^{\left(n\right)}\right)}^T$ of ${}_{\overline{\text{Ω}}}^{xt}$, the space-time domain of $n$-th space-time strip.

Since $2m_1$ and $2m_2$ are the highest orders of the derivatives of $\varphi$ in the space and time, then we have:

${\varphi }_h\left(x,t\right)\in V\subset H^{\left(k,p\right)}\left({\left({\bar{\Omega }}_{xt}^{\left(n\right)}\right)}^T\right)$(4)

in which

$\begin{array}{l}{k}_1\ge 2m_1+1,k_2\ge 2m_2+1,\hfill \\ p_1\ge 2k_1-1,p_2\ge 2k_2-1\hfill \end{array}$(5)

Thus,

${\varphi }_h^e\left(x,t\right)\in V\subset H^{\left(k,p\right)}\left({\bar{\Omega }}_{xt}^{\left(e\right)}\right)$(6)

must hold i.e. the union of local approximation ${\varphi }_h^e$ in (3) must yield the desired global differentiability of ${\varphi }_h\left(x,t\right)$ over ${\left({\bar{\Omega }}_{xt}^{\left(n\right)}\right)}^T$. Equation (6) implies that the local approximation function $N_i\left(.,.\right)$ must belong to the scalar product space $H^{\left(k,p\right)}\left({\Omega }_{xt}^e\right)$ defined by (5), in which

$k_1=2m_1+1,k_2=2m_2+1\text{and}{p}_1=2k_1-1,p_2=2k_2-1$

define minimally conforming scalar product approximation space for which the space-time integrals over ${\left({\bar{\Omega }}_{xt}^{\left(n\right)}\right)}^T$ will be Riemann.

In the space-time finite element method based on space-time residual functional, the space-time integral forms that are space-time variationally consistent for linear as well as non-linear space-time differential operators [1] , hence are meritorious for obtaining solutions of initial value problems. Details of space-time coupled finite element methods for linear and nonlinear differential operators can be found in [1] and are omitted here for the sake of brevity.

If ${\varphi }_h\left(x,t\right)$ in (4) is minimally conforming space or is of class higher than minimally conforming, then the space-time integrals over ${\left({\Omega }_{xt}^{\left(n\right)}\right)}^T$ are Riemann and the residual function over ${\left({\Omega }_{xt}^{\left(n\right)}\right)}^T$ and is given by:

$A{\varphi }_h-f=E\left(x,t\right)\forall \left(x,t\right)\in {\left({\Omega }_{xt}^{\left(n\right)}\right)}^T$(7)

residual functional $I\left({\varphi }_h\right)$ is given by

$I\left({\varphi }_h\right)=\left(E,E\right)$(8)

or

$I\left({\varphi }_h\right)=\underset{e}{{\displaystyle \sum }}{I}^e\left({\varphi }_h^e\right)=\underset{e}{{\displaystyle \sum }}{\left(E^e,E^e\right)}_{{\Omega }_{xt}^e}$(9)

in which

$E^e=A{\varphi }_h^e\left(x,t\right)-f\forall x\in {\bar{\Omega }}_{xt}^e$(10)

and $I^e$ is the residual functional and $E^e$ is the residual function over ${\bar{\Omega }}_{xt}^e$, domain a space-time element e.

For a space-time strip we calculate $I^e$ and $I$ using (9). When ${\varphi }_h\to \varphi$ , the theoretical solution, then

$I\left({\varphi }_h\right)\to I\left(\varphi \right)=0$(11)

Thus, $I^e\left({\varphi }_h\right)$ is a posteriori measure of error in space-time element $e$ and $I\left({\varphi }_h\right)$ is the a posteriori measure of error for the space-time discretization of a space-time strip.

The procedure for calculating a posteriori error based on space-time residual functional for space-time coupled finite element methods for initial value problem is given in the following:

1) Choose minimally conforming spaces for local approximation based on the highest orders of the derivatives of dependent variable(s) in space and time so that the space-time integral over the space-time finite element discretization is Riemann. If the evolution is sufficiently smooth, the order of the space in space and time can be lower by one, if the space-time integral in Lebesgue sense suffices over the space-time discretization depending on the smoothness of the solution.

2) Calculate the solution for the first space-time strip or slab, i.e., ${\varphi }_h^e$ and ${\varphi }_h={\displaystyle {\cup }_e{\varphi }_h^e}$. In cases of nonlinear initial value problems, calculate the converged solution using iterative solution methods (such as Newton’s linear method).

3) Using ${\varphi }_h^e\left(x,t\right)$ for each space-time element, calculate the residual functionals $I^e\left({\varphi }_h^e\right)$ for ${\bar{\Omega }}_{xt}^e$ and $I\left({\varphi }_h\right)$ for ${\left({\bar{\Omega }}_{xt}^{\left(1\right)}\right)}^T$ space-time discretization of the first space-time strip.

$I^e\left({\varphi }_h^e\right)={\left(E^e,E^e\right)}_{{\bar{\Omega }}_{xt}^e},E^e=A{\varphi }_h^e-f$(12)

$I\left({\varphi }_h\right)=\underset{e}{{\displaystyle \sum }}{I}^e\left({\varphi }_h^e\right)$(13)

$I\left({\varphi }_h\right)$ is a measure of error in the computed solution for the first space-time strip or slab.

4) When ${\varphi }_h\to \varphi$, $I\left({\varphi }_h\right)\to I\left(\varphi \right)=0$, thus proximity of calculated $I\left({\varphi }_h\right)$ to computed zero ( $\mathcal{O}\left({10}^{-4}\right)$ or lower) is a measure of error in the solution for the space-time discretization of ${\bar{\Omega }}_{xt}^{\left(1\right)}$ and $I^e\left({\varphi }_h^e\right)$ is a measure of error in the space-time finite element with space-time domain ${\bar{\Omega }}_{xt}^e$.

5) We use $h$, $p$ adaptive refinement for ${\bar{\Omega }}_{xt}^{\left(1\right)}$ to obtain converged solutions (i.e., $I\left({\varphi }_h\right)\le \mathcal{O}\left({10}^{-4}\right)$), keeping minimally conforming $k=\left(k_1,k_2\right)$. $I^e\left({\varphi }_h^e\right)$ are helpful in refining mesh or changing $p$-levels in space and time. Upon obtaining converged solutions for ${\bar{\Omega }}_{xt}^{\left(1\right)}$, we time march to the next space-time strip $\left(t_0+\text{Δ}t\le t\le t_0+2\text{Δ}t\right)$, keeping $h$, $p$ and $k$ fixed. In general, this procedure works well. If a change in $h$ or a change in $p$-level is necessary for subsequent space-time strips, then, unfortunately, we must again begin with ${\bar{\Omega }}_{xt}^{\left(1\right)}$ with the new mesh and $p$-levels.

6) We note that for a space-time strip or a space-time slab the residual functional $I^e$ for an element $e$ with domain ${\bar{\Omega }}_{xt}^e$ is a measure of the error in the solution for element e. These residual functionals $I^e,e=1,2,\cdots$ are quite effective in designing an $h,p$ (possibly $k$) adaptive strategy. The element with the largest value of $I^e$ is the first candidate for $h$ or $p$ or $hp$-refinements. This process can be designed such that all elements with a value of $I^e$ larger than $I^{\text{*}}$ (a threshold value) are considered for $h$ or $p$ or $hp$-refinements. How these refinements are carried out is beyond the scope of this work. Here we only wish to point out that quantities $I^e$ calculated in ‘a posteriori’ error computations indeed provide excellent measures to design adaptive processes.

Let

$A\varphi -f=0\text{in}{\Omega }_{xt}={\Omega }_x\times {\Omega }_t$(14)

be the initial value problem over the space-time domain ${\Omega }_{xt}$, in which $A$ is a space-time differential operator of the highest order $2m_1$ in space and $2m_2$ in time. ${\bar{\Omega }}_{xt}={\Omega }_{xt}\cup \Gamma$, where $\text{Γ}$ is the closed boundary of ${\Omega }_{xt}$. We discretize the spatial domain ${\bar{\Omega }}_x$ with finite elements in ${ℝ}^1$, or in ${ℝ}^2$ or ${ℝ}^3$ in space such that ${\bar{\Omega }}_x^T$ is the discretization of ${\bar{\Omega }}_x$, and consists of ${\bar{\Omega }}_x^T={\displaystyle \underset{e}{\cup }{\bar{\Omega }}_x^e}$, in which ${}^{\overline{\text{Ω}}}\left(x\right)$ is a finite element in the spatial domain ${\bar{\Omega }}_x$. Consider an increment of time $\text{Δ}t$, and let ${\varphi }_h\left(x,t\right)$ be the approximation of $\varphi$ over the spatial domain ${\bar{\Omega }}_x^T={\displaystyle \underset{e}{\cup }{\bar{\Omega }}_x^e}$ for time $t_0\le t\le t_0+\text{Δ}t$ in which $t=t_0$ is the initial time when the evolution described by (14) commences. We consider ${\varphi }_h\left(x,t\right)$ to be the union of ${\varphi }_h^e\left(x,t\right)$

${\varphi }_h\left(x,t\right)={\displaystyle \underset{e}{\cup }{\varphi }_h^e\left(x,t\right)}$(15)

in which ${\varphi }_h^e\left(x,t\right)$ is the approximation of $\varphi \left(x,t\right)$ over ${\bar{\Omega }}_x^e$ for time $t_0\le t\le t_0+\text{Δ}t$, given by:

${\varphi }_h^e\left(x,t\right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{N}_i\left(x\right){\delta }_i^e\left(t\right)\text{and}{\varphi }_h\left(x,t\right)={\displaystyle \underset{e}{\cup }{\varphi }_h^e\left(x,t\right)}$(16)

In (16), $N_i\left(x\right)$ are the approximation functions that are functions of space coordinates only and ${\delta }_i^e\left(t\right)$ are nodal degrees of freedom that are a functions of time only. The local approximation ${\varphi }_h^e\left(x,t\right)$ (16) decouples space and time for ${\bar{\Omega }}_{xt}^e$ and ${\varphi }_h\left(x,t\right)$ decouples space and time for the discretization ${\left({\bar{\Omega }}_x\right)}^T\times {\bar{\Omega }}_t={\left({\bar{\Omega }}_x\right)}^T\times \left[t_0,t_0+\Delta t\right]$. Using fundamental lemma of the calculus of variations, we construct the integral form of (14) over the discretization ${\bar{\Omega }}_x^T$ of the spatial domain ${\bar{\Omega }}_x$ using a test function $v\left(x\right)$ and treating time as a constant.

${\left(A{\varphi }_h\left(x,t\right)-f,v\left(x\right)\right)}_{{\left({\bar{\Omega }}_x\right)}^T}=0$(17)

Since (17) is a functional, we can write this as a sum of the functional over ${\bar{\Omega }}_x^e$ i.e.

${\left(A{\varphi }_h\left(x,t\right)-f,v\right)}_{{\bar{\Omega }}_x^T}=\underset{e}{{\displaystyle \sum }}{\left(A{\varphi }_h^e\left(x,t\right)-f,v\right)}_{{\bar{\Omega }}_x^e}=0$(18)

Consider $\left(A{\varphi }_h^e\left(x,t\right)-f,v\left(x\right)\right)$ in Galerkin methods with weak form in space treating time constant, we perform integration by parts in space if beneficial [1] [9] to obtain the following:

${\left(A{\varphi }_h^e\left(x,t\right)-f,v\left(x\right)\right)}_{{\bar{\Omega }}_x^T}\underset{\text{IBP}}{=}{B}^e\left({\varphi }_h^e,v\right)-l^e\left(v\right)$(19)

We perform some integration by parts in ${\left(A{\varphi }_h^e\left(x,t\right)-f,v\right)}_{{}_{\overline{\text{Ω}}}^x}$ only for those terms that contain even order derivatives in spatial coordinate $x$ to transfer half of the differentiation from ${\varphi }_h^e\left(x,t\right)$ to the test function $v\left(x\right)$. Integral form (19) for ${\bar{\Omega }}_x^e$ is called the weak form of the initial value problem (14) in space. In the weak form the order of differentiation on ${\varphi }_h^e\left(x,t\right)$, with respect to $x$ may have been lowered or weakened, hence the name weak form, but increased on the test function. After performing the integration in space for an element $e$, we obtain a system of ordinary differential equations in time and $\left\{{\delta }^e\right\}$ and its time derivatives. The order of the time derivatives of $\left\{{\delta }^e\right\}$ in the ordinary differential equations depends upon the order of the time derivatives of $\varphi$ in (14), that is (19) can be written as

$\begin{array}{l}{\left(A{\varphi }_h^e\left(x,t\right),v\left(x\right)\right)}_{{\bar{\Omega }}_x^e}=B^e\left({\varphi }_h^e,v\left(x\right)\right)-l^e\left(v\right)\\ =H_1^e\left\{{\delta }^e\right\}+H_2^e\left\{{\stackrel{\dot{}}{\delta }}^e\right\}+\cdots +H_n^e\left\{{\delta }_{\left[n\right]}^e\right\}-\left\{P^e\right\}-\left\{F^e\right\}\end{array}$(20)

Substituting (20) into (18), we obtain

${\left(A{\varphi }_h\left(x,t\right)-f,v\right)}_{{\bar{\Omega }}_{xt}^T}=\left[H_1\left]\left\{\delta \right\}+\right[H_2\right]\left\{\stackrel{\dot{}}{\delta }\right\}+\cdots +\left[H_n\right]\left\{{\delta }_{\left[n\right]}\right\}-\left\{P\right\}-\left\{F\right\}=0$ (21)

In which

$H_1=\underset{e}{{\displaystyle \sum }}\left[H_1^e\right]$, $H_2=\underset{e}{{\displaystyle \sum }}\left[H_2^e\right]$… etc. and $\left\{\delta \right\}={\displaystyle \underset{e}{\cup }\left\{{\delta }_e\right\}}$, $\left\{\stackrel{\dot{}}{\delta }\right\}={\displaystyle \underset{e}{\cup }\left\{{\stackrel{\dot{}}{\delta }}_e\right\}}$… etc.

In (21), we have a system of ordinary differential equations in time for the initial value problem (14) for ${\bar{\Omega }}_{xt}^T={\bar{\Omega }}_x^T\times \left[t_0,t_0+\Delta t\right]$

The system of ordinary differential equations in (21) is linear if the space-time differential operator $A$ in (14) is linear, otherwise (21) represents a system of non-linear ordinary differential equations in time.

The ordinary differential equations in time in (21) can be integrated in time using:

1) Finite element methods in time

2) Explicit time integration methods

3) Implicit time integration methods

Explicit time integration methods have poor accuracy and stability issues but are much simpler and are quite fast computationally. We do not consider these methods in this work because of poor accuracy, but also due to other requirements discussed later.

Consider space-time coupled finite element method for initial value problems. The local approximation of ${\varphi }^e\left(x,t\right)$ over a space-time element ${\bar{\Omega }}_x^e$ or ${\bar{\Omega }}_{\xi }^e$ is given by:

${\varphi }_h^e\left(x,t\right)=\left[N\left(x,t\right)\right]\left\{{\delta }^e\right\}$(22)

If $\left[N\left(x,t\right)\right]$ are generated using tensor product, then we can write (22) as follows:

${\varphi }_h^e\left(x,t\right)=\left[N\left(x\right)\right]\otimes \left[N\left(t\right)\right]\left\{{\delta }^e\right\}$(23)

The $\otimes$ symbol indicates tensor product [1] [6] as we use it in finite elements to generate 2D approximation functions from 1D approximation functions.

We can also write (23) as:

${\varphi }_h^e\left(x,t\right)=\left[N\left(x\right)\right]\left(\left[N\left(t\right)\right]{\delta }^e\right)$(24)

or

${\varphi }_h^e\left(x,t\right)=\left[N\left(x\right)\right]\left\{{\delta }^e\left(t\right)\right\}$(25)

where

${\delta }^e\left(t\right)=\left[N\left(t\right)\right]\left\{{\delta }^e\right\}$(26)

In (24) and (26) $\left\{{\delta }^e\right\}$ is not a function of time.

Remarks

1) Equation (24) is the space-time coupled finite element method.

2) Equation (25) is the space-time decoupled finite element method.

3) Equation (26) is the finite element method in time.

4) Substitution from (26) into (25) yields (24).

That is in space-time decoupled methods, if we solve ordinary differential equations in time using finite element method in time, then the space-time coupled and the space-time decoupled methods are the same provided the space-time approximation functions are generated using tensor product of 1D space and time approximation functions. This has been investigated by Surana et al. [12] and is shown to be true. For further details on this approach, see reference [12] .

The a posteriori error computation proposed in the present work considers $L_2$-norm of the space-time residual function for space-time decoupled finite element methods. This technique requires:

(i) Desired regularity or differentiability of ${\varphi }_h^e\left(x,t\right)=\left[N\left(x\right)\right]{\delta }^e\left(t\right)$ in space and time. Since the spatial discretization uses finite elements in space, the desired regularity in space can be easily satisfied by appropriate choices of $k$ and $p$ in space.

(ii) We also need desired global regularity or differentiability of ${\varphi }_h\left(x,t\right)$ in time as well. That is, we must consider a time integration method in which approximation of $\varphi$ over a time increment is a function or an expression in time (for example, polynomial in time).

(iii) At the inter-element boundaries, desired order of differentiability required by the initial value problem must be satisfied so that the space-time integrals are Riemann.

(iv) When the requirements (i) - (iii) are satisfied, the space-time decoupled approximation ${\varphi }_h^e\left(x,t\right)=\left[N\left(x\right)\right]\left\{{\delta }^e\left(t\right)\right\}$ can be substituted in the initial value problem (14) for any value of time in the interval $\left(n-1\right)\text{Δ}t\le t\le n\text{Δ}t$. Thus, permitting $L_2$-norm of the space-time residual functional over ${\bar{\Omega }}_x^e\times \left[\left(n-1\right)\Delta t\le t\le n\Delta t\right]$ for the $n^{th}$ space-time step.

(v) The requirements (i) - (iv) rule out the use of explicit time integration methods and the integration method in which we do not have an expression of the solution as a function of time explicitly for an increment of time.

Remarks [a.]

a. Requirements (i) - (iv) rule out methods like Runge-Kutta in which we do not have an analytical expression of solution as a function of time in a time interval as required in the $L_2$-norm computations.

b. Wilson’s $\theta$-method with linear acceleration and Newmark’s linear acceleration methods are suitable for second-order systems of ordinary differential equations in time.

c. In the case of first-order systems of ordinary differential equations in time, we must derive details of time integration methods based on Wilson’s $\theta$ method and Newmark method.

d. In the present work, we consider the Newmark method for both first-order and second-order systems of ordinary differential equations in time. For simplicity, we only consider linear ordinary differential equations in time. In the following, we present details of Newmark time integration method for linear first-order and second-order ordinary differential equations in time.

Consider the following first-order system of linear ordinary differential equations in time obtained from decoupling of space and time

$\left[C\right]\left\{\stackrel{\dot{}}{\delta }\right\}+\left[K\right]\left\{\delta \right\}=\left[F\right]$(27)

Consider a time interval $\left[t,t+\text{Δ}t\right]$ and assume that $\left\{\stackrel{\dot{}}{\delta }\right\}$ is linear in the interval $\left[t,t+\text{Δ}t\right]$ ( Figure 2 ).

Let $\tau$ be the time measured from $t$. Let ${\left\{\stackrel{\dot{}}{\delta }\right\}}_t$ and ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$ be values of $\left\{\stackrel{\dot{}}{\delta }\right\}$ at $t$ and $t+\text{Δ}t$, respectively. Then

${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\tau }={\left\{\stackrel{\dot{}}{\delta }\right\}}_t+\frac{\tau }{\text{Δ}t}\left({\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}-{\left\{\stackrel{\dot{}}{\delta }\right\}}_t\right)$(28)

Integrating with respect to $\tau$, we get:

${\left\{\delta \right\}}_{t+\tau }=\tau {\left\{\stackrel{\dot{}}{\delta }\right\}}_t+\frac{{\tau }^2}{2\text{Δ}t}\left({\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}-{\left\{\stackrel{\dot{}}{\delta }\right\}}_t\right)+C$(29)

At $\tau =0$,

${\left\{\delta \right\}}_{t+\tau }={\left\{\delta \right\}}_t$(30)

Substituting in (29), we obtain:

$C={\left\{\delta \right\}}_t$(31)

$\therefore {\left\{\delta \right\}}_{t+\tau }={\left\{\delta \right\}}_t+\tau {\left\{\stackrel{\dot{}}{\delta }\right\}}_t+\frac{{\tau }^2}{2\text{Δ}t}\left({\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}-{\left\{\stackrel{\dot{}}{\delta }\right\}}_t\right)$(32)

Letting $\tau =\text{Δ}t$, we obtain:

${\left\{\delta \right\}}_{t+\text{Δ}t}={\left\{\delta \right\}}_t+\text{Δ}t{\left\{\stackrel{\dot{}}{\delta }\right\}}_t+\frac{\text{Δ}t}{2}\left({\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}-{\left\{\stackrel{\dot{}}{\delta }\right\}}_t\right)$(33)

From (28),

${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}={\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$

From (33), we solve for ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$:

${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}=\frac{2}{\text{Δ}t}\left({\left\{\delta \right\}}_{t+\text{Δ}t}-{\left\{\delta \right\}}_t\right)-{\left\{\stackrel{\dot{}}{\delta }\right\}}_t$(34)

Consider (27) at $t+\text{Δ}t$:

$\left[C\right]{\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}+\left[K\right]{\left\{\delta \right\}}_{t+\text{Δ}t}={\left\{F\right\}}_{t+\text{Δ}t}$(35)

Substitute ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$ from (34) into (35):

$\left[C\right]\left(\frac{2}{\text{Δ}t}\left({\left\{\delta \right\}}_{t+\text{Δ}t}-{\left\{\delta \right\}}_t\right)-{\left\{\stackrel{\dot{}}{\delta }\right\}}_t\right)+\left[K\right]{\left\{\delta \right\}}_{t+\text{Δ}t}={\left\{F\right\}}_{t+\text{Δ}t}$(36)

or

$\left(\left[K\right]+\frac{2}{\text{Δ}t}\left[C\right]\right){\left\{\delta \right\}}_{t+\text{Δ}t}={\left\{F\right\}}_{t+\text{Δ}t}+\frac{2}{\text{Δ}t}\left[C\right]{\left\{\delta \right\}}_t-\left[C\right]{\left\{\stackrel{\dot{}}{\delta }\right\}}_t$(37)

or

$\left[B\right]{\left\{\delta \right\}}_{t+\text{Δ}t}={\left\{Q\right\}}_t+{\left\{F\right\}}_{t+\text{Δ}t}$(38)

where

${\left\{Q\right\}}_t=\frac{2}{\text{Δ}t}\left[C\right]{\left\{\delta \right\}}_t-\left[C\right]{\left\{\stackrel{\dot{}}{\delta }\right\}}_t$(39)

$\left[B\right]$ is called the time approximation operator, and ${\left\{Q\right\}}_t$ is called the load operator.

Knowing ${\left\{\delta \right\}}_t$, ${\left\{\stackrel{\dot{}}{\delta }\right\}}_t$, and ${\left\{F\right\}}_{t+\text{Δ}t}$, we can calculate ${\left\{\delta \right\}}_{t+\text{Δ}t}$ using (37), and then ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$ using (34). Thus, the solution ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$ and ${\left\{\delta \right\}}_{t+\text{Δ}t}$ is now known.

Initial conditions

1) Since (27) is a first-order system of ordinary differential equations, we require only one initial condition. If we begin with $t=0$ then ${\left\{\delta \right\}}_0$ is the initial condition (known). Using (27) at $t=0$, we can write:

$\left[C\right]{\left\{\stackrel{\dot{}}{\delta }\right\}}_0+\left[K\right]{\left\{\delta \right\}}_0={\left\{F\right\}}_0$(40)

From (27), we can solve for ${\left\{\stackrel{\dot{}}{\delta }\right\}}_0$:

${\left\{\stackrel{\dot{}}{\delta }\right\}}_0={\left[C\right]}^{-1}\left(-\left[K\right]{\left\{\delta \right\}}_0+{\left\{F\right\}}_0\right)$(41)

Thus, now the complete solution is known at $t=0$.

2) Using the initial condition in (1) at $t=0$, we can use (34) at $t=0$ to calculate ${\left\{\delta \right\}}_{t+\text{Δ}t}$ and then use (33) to calculate ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$

3) This procedure of time marching is continued until final time $t=\tau$ is reached.

Remarks

1) First, we note that for each time interval, we have an analytical expression for $\left\{\delta \left(t\right)\right\}$ and $\left\{\stackrel{\dot{}}{\delta }\left(t\right)\right\}$ needed for residual computations.

2) Secondly, we note that at a time $t$ between the two time intervals, $\left\{\delta \right\}$ and $\left\{\stackrel{\dot{}}{\delta }\right\}$ are continuous, implying that in this time integration scheme, $\delta \left(t\right)$ is of class $C^1$ as required by (14) to ensure that the time integrals in $L_2$-norm are Riemann in time.

Consider a system of second-order linear ordinary differential equations in time:

$\left[M\right]\left\{\ddot{\delta }\right\}+\left[C\right]\left\{\stackrel{\dot{}}{\delta }\right\}+\left[K\right]\left\{\delta \right\}=\left\{F\right\}$(42)

These ordinary differential equations in time result when decoupling space and time in balance of linear momenta in linear elasticity and when constructing integral forms over spatial discretization using Galerkin method with weak form (GM/WF). $\left[M\right]$, $\left[C\right]$, and $\left[K\right]$ are mass, damping, and stiffness matrices, and $\left\{\delta \right\}$, $\left\{\stackrel{\dot{}}{\delta }\right\}$, and $\left\{\ddot{\delta }\right\}$ are nodal degrees of freedom, their first and second time derivatives, when the degrees of freedom $\left\{\delta \right\}$ are displacements (C0 Lagrange family finite elements), then $\left\{\stackrel{\dot{}}{\delta }\right\}$ and $\left\{\ddot{\delta }\right\}$ are velocities and accelerations of the grid points of the spatial discretization.

We consider Newmark’s linear acceleration method for integrating ordinary differential Equations (42) in time. In this method, we assume that acceleration $\left\{\ddot{\delta }\right\}$ is linear in the time interval $\left[t,t+\text{Δ}t\right]$ ( Figure 3 ).

Complete details of this method can be found in [1] . Important steps and the calculation procedure are described in the following. If we measure time $\tau$ from $t$, then

${\left\{\ddot{\delta }\right\}}_{t+\tau }={\left\{\ddot{\delta }\right\}}_t+\frac{\tau }{\text{Δ}t}\left({\left\{\ddot{\delta }\right\}}_{t+\text{Δ}t}-{\left\{\ddot{\delta }\right\}}_t\right)$(43)

Integrating with respect to $\tau$ and using $\left\{\stackrel{\dot{}}{\delta }\right\}$ at $\tau =0$ equal as ${\left\{\stackrel{\dot{}}{\delta }\right\}}_t$, to evaluate constant of integration, we obtain

${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\tau }=\tau {\left\{\ddot{\delta }\right\}}_t+{\left\{\stackrel{\dot{}}{\delta }\right\}}_t+\frac{{\tau }^2}{2\text{Δ}t}\left({\left\{\ddot{\delta }\right\}}_{t+\text{Δ}t}-{\left\{\ddot{\delta }\right\}}_t\right)$(44)

Integrating with respect to $\tau$ and using ${\left\{\delta \right\}}_{t+\tau }$ at $\tau =0$, as ${\left\{\delta \right\}}_t$ to evaluate the constant of integration, we obtain

${\left\{\delta \right\}}_{t+\tau }={\left\{\delta \right\}}_t+\frac{{\tau }^2}{2}{\left\{\ddot{\delta }\right\}}_t+\frac{{\tau }^3}{6\text{Δ}t}\left({\left\{\ddot{\delta }\right\}}_{t+\text{Δ}t}-{\left\{\ddot{\delta }\right\}}_t\right)$(45)

Using (43), (44), and (45), we obtain expressions for ${\left\{\delta \right\}}_{t+\text{Δ}t}$, ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$ and ${\left\{\ddot{\delta }\right\}}_{t+\text{Δ}t}$ by letting $\tau =\text{Δ}t$. From these, we solve for ${\left\{\ddot{\delta }\right\}}_{t+\text{Δ}t}$ and ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$ in terms of ${\left\{\delta \right\}}_t$, ${\left\{\stackrel{\dot{}}{\delta }\right\}}_t$, ${\left\{\ddot{\delta }\right\}}_t$, ${\left\{\delta \right\}}_{t+\text{Δ}t}$ and ${\left\{F\right\}}_{t+\text{Δ}t}$ and substitute these into (43) at $t+\text{Δ}t$.

$\left[M\right]{\left\{\ddot{\delta }\right\}}_{t+\text{Δ}t}+\left[C\right]{\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}+\left[K\right]{\left\{\delta \right\}}_{t+\text{Δ}t}={\left\{F\right\}}_{t+\text{Δ}t}$(46)

We express resulting (46) as a system of linear simultaneous algebraic equations in unknown ${\left\{\delta \right\}}_{t+\text{Δ}t}$ (see [1] for details), Equation (47). Final expression used for computations are given in the following:

$\begin{array}{l}\left(\frac{6}{\text{Δ}{t}^2}\left[M\right]+\frac{3}{\text{Δ}t}\left[C\right]+\left[K\right]\right){\left\{\delta \right\}}_{t+\text{Δ}t}\\ ={\left\{F\right\}}_{t+\text{Δ}t}+\left(\frac{6}{\text{Δ}{t}^2}\left[M\right]+\frac{3}{\text{Δ}t}\left[C\right]\right){\left\{\delta \right\}}_t+\left(\frac{6}{\text{Δ}t}\left[M\right]+2\left[C\right]\right){\left\{\stackrel{\dot{}}{\delta }\right\}}_t\\ +\left(2\left[M\right]+\frac{\text{Δ}t}{2}\left[C\right]\right){\left\{\ddot{\delta }\right\}}_t\end{array}$(47)

and

${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}=\frac{3}{\text{Δ}t}\left({\left\{\delta \right\}}_{t+\text{Δ}t}-{\left\{\delta \right\}}_t\right)-2{\left\{\ddot{\delta }\right\}}_t-\frac{\text{Δ}t}{2}{\left\{\ddot{\delta }\right\}}_t$(48)

${\left\{\ddot{\delta }\right\}}_{t+\text{Δ}t}=\frac{6}{\text{Δ}{t}^2}\left({\left\{\delta \right\}}_{t+\text{Δ}t}-{\left\{\delta \right\}}_t\right)-\frac{6}{\text{Δ}t}{\left\{\stackrel{\dot{}}{\delta }\right\}}_t-2{\left\{\ddot{\delta }\right\}}_t$(49)

We can write (47) as

$\left[B\right]{\left\{\delta \right\}}_{t+\text{Δ}t}={\left\{Q\right\}}_t+{\left\{F\right\}}_{t+\text{Δ}t}$

where

$\begin{array}{c}{\left\{Q\right\}}_t=\left(\frac{6}{\text{Δ}{t}^2}\left[M\right]+\frac{3}{\text{Δ}t}\left[C\right]\right){\left\{\delta \right\}}_t+\left(\frac{6}{\text{Δ}t}\left[M\right]+2\left[C\right]\right){\left\{\stackrel{\dot{}}{\delta }\right\}}_t\\ +\left(2\left[M\right]+\frac{\text{Δ}t}{2}\left[C\right]\right){\left\{\ddot{\delta }\right\}}_t\end{array}$(51)

Initial Conditions:

Since (42) is a system of linear second-order ordinary differential equations in time, it requires two initial conditions. If we assume that the evolution commences at $t=0$, then we could have ${\left\{\delta \right\}}_0$ and ${\left\{\stackrel{\dot{}}{\delta }\right\}}_0$ as initial conditions. Using (42) at $t=0$, $\left[M\right]{\left\{\ddot{\delta }\right\}}_0+\left[C\right]{\left\{\stackrel{\dot{}}{\delta }\right\}}_0+\left[K\right]{\left\{\delta \right\}}_0={\left\{F\right\}}_0$(52)

From (52), we can determine ${\left\{\ddot{\delta }\right\}}_0$.

${\left\{\ddot{\delta }\right\}}_0={\left[M\right]}^{-1}\left({\left\{f\right\}}_0-\left[C\right]{\left\{\stackrel{\dot{}}{\delta }\right\}}_0-\left[K\right]\left\{{\delta }_0\right\}\right)$

Thus, ${\left\{\delta \right\}}_0$, ${\left\{\stackrel{\dot{}}{\delta }\right\}}_0$, and ${\left\{\ddot{\delta }\right\}}_0$ are all known at time $t_0=0$.

Remarks

1) If the evolution commences at $t=0$, then ${\left\{\delta \right\}}_0$, ${\left\{\stackrel{\dot{}}{\delta }\right\}}_0$, and ${\left\{\ddot{\delta }\right\}}_0$ are all known at $t=0$.

2) We use $t=0$ in (45) to calculate ${\left\{\delta \right\}}_{\text{Δ}t}$ followed by the calculation of ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\text{Δ}t}$ and ${\left\{\ddot{\delta }\right\}}_{t+\text{Δ}t}$ using (48) and (49). Thus, the solution at $t=\text{Δ}t$ is now known.

3) Using the solution at $t=\text{Δ}t$, the solution at $t=2\text{Δ}t$ i.e. ${\left\{\delta \right\}}_{2\text{Δ}t}$, ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{2\text{Δ}t}$, and ${\left\{\ddot{\delta }\right\}}_{2\text{Δ}t}$ can be calculated using (47) - (49).

4) This time marching process is continued until the final time $t=\tau$ is reached.

Consider the following system of linear first order ordinary differential equations in time:

$\left[C\right]\left\{\stackrel{\dot{}}{\delta }\right\}+\left[H\right]\left\{\delta \right\}=\left\{F\right\}$(53)

Obtained by decoupling space and time in the initial value problem using a finite element discretization ${\left({\bar{\Omega }}_{xt}\right)}^T={\displaystyle \underset{e}{\cup }{\bar{\Omega }}_x^e}$ in space of the spatial domain ${\bar{\Omega }}_x\subset {ℝ}^1$ or ${ℝ}^2$ or ${ℝ}^3$; ${\bar{\Omega }}_x^e$ could be a line, an area or a volume finite element in space,with p-version hierarchical with higher orders global differentiability in space.

For illustration, consider the following time dependent diffusion equation

$c\frac{\partial \varphi }{\partial t}-k\frac{{\partial }^2\varphi }{\partial x^2}-f=0,\forall x,t\in {\Omega }_x\times {\Omega }_t=\left(0,1\right)\times \left(0,\tau \right)$(54)

Consider discretization of ${\bar{\Omega }}_x^T$ of the spatial domain ${\bar{\Omega }}_x$

${\bar{\Omega }}_x^T={\displaystyle \underset{e}{\cup }{\bar{\Omega }}_x^e}$(55)

${\bar{\Omega }}_x^e$ is a three-node p-version hierarchical finite element in the spatial domain $x$ with higher-order global differentiability local approximation. Let ${\varphi }_h^e\left(x,t\right)$ be the approximation of $\varphi$ over ${\bar{\Omega }}_x^e$, then in space-time decoupled finite element method, we write:

${\varphi }_h^e\left(x,t\right)=\left[N\left(x\right)\right]\left\{{\delta }^e\left(t\right)\right\}$(56)

in which ${\delta }^e\left(t\right)$ is the computed solution for time integration method for an increment of time.

We substitute (56) in (54) to obtain residual function $E_{xt}^e$. ${\underset{˜}{\bar{\Omega }}}_{xt}^e={\bar{\Omega }}_x^e\times \left[\left(n-1\right)t\le t\le nt\right]$, for the $n^{th}$ time interval

$E_{xt}^e\left(x,t\right)=C\left[N\left(x\right)\right]\left\{\stackrel{\dot{}}{\delta }\left(t\right)\right\}-k\left[\frac{{\partial }^{2}N\left(x\right)}{\partial x^2}\right]\left\{\delta \left(t\right)\right\}-f\left(t\right)$(57)

and the residual functional $I_{xt}^e$ is given by

$I_{xt}^e={\left(E_{xt}^e\left(x,t\right),E_{xt}^e\left(x,t\right)\right)}_{{\underset{˜}{\bar{\Omega }}}_{xt}^e}$(58)

and for the the whole spatial discretization

$I_{xt}=\underset{e}{{\displaystyle \sum }}{I}_{xt}^e$(59)

We note that $N\left(x\right)$ are approximation functions for ${\bar{\Omega }}_x^e$ mapped in natural coordinate space $\xi$. Thus,

$E_{\xi t}^e\left(\xi ,t\right)=C\left(\left[N\left(\xi \right)\right]\right)\left\{\stackrel{\dot{}}{\delta }\left(t\right)\right\}+K\left[\frac{{\partial }^{2}N\left(\xi \right)}{\partial x^2}\right]\left\{\delta \left(t\right)\right\}-f\left(t\right)$(60)

$I_{\xi t}=\underset{e}{{\displaystyle \sum }}{I}_{\xi t}^e=\underset{e}{{\displaystyle \sum }}{\left(E_{\xi t}^e\left(\xi ,t\right),E_{\xi t}^e\left(\xi ,t\right)\right)}_{{\underset{˜}{\bar{\Omega }}}_{\xi t}^e}$(61)

We map time $t$ for an increment of time into natural coordinate $\nu$ in a two-unit length. Consider the $j^{th}$ time increment $\left[t^j,t^j+\text{Δ}t\right]=\left[t_1^j,t_2^j\right]$ mapped into a two-unit length $\left[{\nu }_1,{\nu }_2\right]=\left[-1,1\right]$. We can write the following:

$t\left(\nu \right)=\left(\frac{1-\nu }{2}\right)t_1^j+\left(\frac{1+\nu }{2}\right)t_2^j$(62)

and

$\tau \left(\nu \right)=\left(\frac{1+\nu }{2}\right)\text{Δ}t\text{and}\text{d}\tau =\frac{\text{Δ}t}{2}\text{d}\nu$(63)

time $\tau$ is measured from $t=t_1^j$.

Recall ${\left\{\delta \right\}}_{t+\tau }$ and ${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t+\tau }$ (from Equations (29) and (28)), we can express these as functions of $\nu$:

${\left\{\delta \right\}}_{t_1+\tau \left(\nu \right)}=\tau \left(\nu \right){\left\{\stackrel{\dot{}}{\delta }\right\}}_{t_1}^j+\frac{{\left(\tau \left(\nu \right)\right)}^2}{2\text{Δ}t}\left({\left\{\stackrel{\dot{}}{\delta }\right\}}_{t_1^j+\text{Δ}t}-{\left\{\stackrel{\dot{}}{\delta }\right\}}_{t_1^j}\right)$(64)

and

${\left\{\stackrel{\dot{}}{\delta }\right\}}_{t_1^j+\tau \left(\nu \right)}={\left\{\stackrel{\dot{}}{\delta }\right\}}_{t_1^j}+\frac{\tau \left(\nu \right)}{\text{Δ}t}\left({\left\{\stackrel{\dot{}}{\delta }\right\}}_{t_1^j+\text{Δ}t}-{\left\{\stackrel{\dot{}}{\delta }\right\}}_{t_1^j}\right)$(65)

where

$t_2^j=t_1^j+\text{Δ}t$(66)

Using (62) in (61) , we obtain $E_{\xi \nu }^e\left(\xi ,\nu \right)$ and:

$I_{\xi \nu }=\underset{e}{{\displaystyle \sum }}{I}_{\xi \nu }^e=\underset{e}{{\displaystyle \sum }}{\left(E_{\left(\xi \nu \right)}^e\left(\xi ,\nu \right),E_{\xi \nu }^e\left(\xi ,\nu \right)\right)}_{{\underset{˜}{\bar{\Omega }}}_{\xi \nu }^e}$(67)

$I_{\xi \nu }^e$ is calculated using Gauss quadrature.

Proximity of $I_{\xi \nu }^e$ to zero is a measure of the accuracy of the solution in ${\underset{˜}{\bar{\Omega }}}_{\xi \nu }^e$ i.e. element e in space for a time increment and proximity of $I_{\xi \nu }$ to zero is a measure of accuracy of the solution over the whole spatial discretization for an increment of time.

In linear elasticity, the balance of linear momenta is a system of second order partial differential equations in space and time in displacement when the constitutive theory for stresses is substituted in them. We discussed space-time decoupled method for obtaining solutions of such initial value problems.

Consider the initial value problem in linear elasticity, axial deformation of rod made of viscoelastic material. The rod has elasticity and dissipation mechanisms. Balance of linear momenta and constitutive theory are given by (neglecting equilibrium stress):

${\rho }_0\frac{{\partial }^{2}u}{\partial t^2}-{\rho }_0{F}_1^b-\frac{\partial {\sigma }_{11}}{\partial x}=0\forall x\in {\Omega }_x\subset {ℝ}^1$(68)

${\sigma }_{11}=E\frac{\partial u}{\partial x}+c\frac{\partial }{\partial t}\left(\frac{\partial u}{\partial x}\right)$(69)

By substituting (69) in (68), we obtain a single partial differential equation in $u$:

${\rho }_0\frac{{\partial }^{2}u}{\partial t^2}-{\rho }_0{F}_1^b-E\frac{{\partial }^{2}u}{\partial x^2}-c\frac{{\partial }^{2}v}{\partial x^2}=0$(70)