**2.3 Discrete analogs of the qualitative properties - reliable discrete models**

**2.3.5 Basic conditions for the finite difference and finite element approx-**

As it was mentioned in Section 2.3.3, the discrete mesh operators are derived via the
discretization of the partial differential operators. Therefore, *L* is usually some
*approx-imation* to *L, which means the following. Let* P denote the projection operator from
the space domL to dom*L* defined as follows. For *v* *∈* dom*L, (Pv)(x**i**, t**n*) = *v(x**i**, t**n*)
(i= 1, . . . ,*N;*¯ *n*= 0, . . . , M). (We note that dom*L, and hence* *L, depends on the choice*

*f* *u*_{0} *u** _{∂}*
DNP non-negative non-negative non-negative

NPCAP non-negative non-negative non-negative and time-decreasing

DWMP any any any

DSMP any any any

DWBMP non-positive any any

Ciarlet non-positive any time-decreasing

DSBMP non-positive any any

Stoyan zero non-negative non-negative and time-independent Table 2.3.1: Conditions for the given data of the continuous problem providing different qualitative properties.

of the mesh ¯*Q*_{t}* _{M}*, i.e., on the discretization parameters

*h*and ∆t. Therefore, for refined meshes they denote a family of the operators.)

Definition 2.3.42 *We say that* *L* *locally approximates the operator* *Lif for all functions*
*v* *∈*dom*L* *and for all points* (x^{?}*, t** ^{?}*)

*∈Q*

*T*

*we have*

(Lv)(x^{?}*, t** ^{?}*)

*−*(L(Pv))(x

^{?}

_{h}*, t*

^{?}_{∆t})

*→*0, (2.3.64)

*when*x

^{?}

_{h}*→*x

^{?}*and*

*t*

^{?}_{∆t}

*→t*

^{?}*as*

*h,*∆t

*→*0.

The expression on the left-hand side in (2.3.64) is called*local approximation error*and
the rate of its convergence to zero defines the order of the approximation. This will be
denoted by the symbol *O(g(∆t, h)), whereg* is some function (typically polynom) of ∆t
and *h.* ^{5}

Aiming at preserving the qualitative properties, we want to use Theorem 2.3.18.

Therefore, first we should analyze validity of the conditions (2.3.12* ^{0}*) and (2.3.13

*). In what follows, we consider the differential operator in the standard form*

^{0}*L≡* *∂*

*∂t* *−*
X*d*

*m=1*

*∂*

*∂x** _{m}*(k

*(x, t)*

_{m}*∂*

*∂x** _{m}*)

*−*X

*d*

*m=1*

*a** _{m}*(x, t)

*∂*

*∂x*_{m}*−a*_{0}(x, t), (2.3.65)
which will be discretized by two popular numerical techniques - the finite difference and
finite element methods. Henceforward we assume that *k** _{m}* are positive functions and

*a*

_{0}is a non-positive function.

a. The finite difference discretization

In the following we approximate the operator *L* in (2.3.65) with sufficiently smooth
co-efficient functions on a rectangular mesh by the usual finite difference method according
to Figure 2.3.2. (See e.g., [73, 116, 133]). The interior points of the mesh are denoted

5The “Big Oh notation” (it is also called as Landau notation, Bachmann-Landau notation, asymptotic
notation) first appeared in the second volume of Bachmann’s treatise on number theory [2], and Landau
obtained this notation in Bachmann’s book [85]. This symbol means the following. Let g*τ* be a vector
function defined on an interval *I ⊂* IR, g*τ* : *I →* IR* ^{n}*, with

*τ*being a scalar parameter. We write g

*τ*(t) =

*O(τ*

*) if there exists a constant*

^{p}*C*0

*>*0 such that for sufficiently small values of

*|τ|*the inequality

*kg**τ*(t)k ≤*C*0*|τ|*^{p}

holds uniformly with respect to*t**∈ I* and*k · k*is any vector norm on IR* ^{n}*.

### x

**x**

i **x**

i + x
**x**

i - x
### h

i + x### h

i - x 1### x

2 11 11Figure 2.3.2: *A grid of a two-dimensional rectangular domain.*

by x_{1}*, . . . ,*x* _{N}* and the boundary ones by x

_{N+1}*, . . . ,*x

*N*¯. For the sake of simplicity, we also use the notation x

_{i+x}*(x*

_{m}

_{i−x}*) for the grid point adjoint to x*

_{m}*in positive (negative)*

_{i}*x*

*m*-direction. The lengths of the segments [x

*i*

*,*x

*i+x*

*m*] and [x

*i−x*

*m*

*,*x

*i*] are denoted by

*h*

*i+x*

*m*

and *h*_{i−x}* _{m}*, respectively. Furthermore, let us denote the uniform temporal discretization
step size with ∆t >0, and we will use the notation

*t*

*=*

_{n}*t*

_{n}*−*0.5∆t. The integer number

*M*is defined by the property

*M*∆t

*≤T <*(M + 1)∆t.

Using the notation *ν*_{i}* ^{n}* for the value of

*v(x*

_{i}*, t*

*), the finite difference approximation results in the discrete mesh operator*

_{n}*L*defined in the canonical form (2.3.17), where for the entries ofM

^{(n)}(=M) we have

*M*_{i,j}^{(n)} =*M** _{i,j}* =
(

1, if *i*=*j*

0, if *i6=j,* *i*= 1, . . . , N; *j* = 1, . . . ,*N.*¯ (2.3.66)
Applying the central difference approximation for the first order derivatives, the nonzero
elements of the *i-th row of* K^{(n)} are *K*_{i,i−x}^{(n)} _{m}*, K*_{i,i+x}^{(n)} * _{m}* (m= 1, . . . , d) and

*K*

_{i,i}^{(n)}, where

*K*_{i,i−x}^{(n)} * _{m}* =

*−2(k*

*)*

_{m}^{(n)}

_{i−0.5}*h*_{i−x}* _{m}*(h

_{i−x}*+*

_{m}*h*

_{i+x}*) + (a*

_{m}*)*

_{m}^{(n)}

_{i}*h*_{i−x}* _{m}*+

*h*

_{i+x}*(2.3.67)*

_{m}*K*

_{i,i+x}^{(n)}

*=*

_{m}*−2(k*

*)*

_{m}^{(n)}

_{i+0.5}*h*_{i+x}* _{m}*(h

_{i−x}*+*

_{m}*h*

_{i+x}*)*

_{m}*−*(a

*)*

_{m}^{(n)}

_{i}*h*_{i−x}* _{m}*+

*h*

_{i+x}*(2.3.68) and*

_{m}*K*_{i,i}^{(n)} =
X*d*

*m=1*

2
*h*_{i−x}* _{m}* +

*h*

_{i+x}

_{m}Ã(k* _{m}*)

^{(n)}

_{i+0.5}*h*_{i+x}* _{m}* +(k

*)*

_{m}^{(n)}

_{i−0.5}*h*

_{i−x}

_{m}!

*−*(a0)^{(n)}_{i}*,* (2.3.69)
where (k*m*)^{(n)}* _{i±0.5}* = 0.5(k

*m*(x

*i*

*, t*

*n*)+k

*m*(x

*i±1*

*, t*

*n*)), (a

*m*)

^{(n)}

*=*

_{i}*a*

*m*(x

*i*

*, t*

*n*) and (a0)

^{(n)}

*=*

_{i}*a*0(x

*i*

*, t*

*n*).

Hence, for the finite difference discrete mesh operators *L, defined by (2.3.17) and*
(2.3.66)–(2.3.69), the relations

Me=e0 (2.3.70)

and

K^{(n)}e
(

*≥*0, if *a*0 *≤*0;

=0, if *a*0 = 0 (2.3.71)

hold. Hence, we have

Theorem 2.3.43 *Let us assume that the finite difference discrete mesh operator* *L, *
*de-fined by* (2.3.17)*and* (2.3.66)-(2.3.69), is non-negativity preserving. Then, beyond the NP
*property, when* *a*0 *≤*0, the operator *L* *is DWMP, DWBMP and DMNC, too; while in the*
*case* *a*_{0} = 0 *it has each of the DWMP, DSMP, DWBMP, DSBMP and DMNC properties.*

Let us replace the central difference approximation with the upwind (upstream) ap-proximation. In this case the matrix M does not change and the elements of K have the following form

respectively. One can directly check that for the finite difference discrete mesh operators
*L, defined by (2.3.17) and (2.3.72)-(2.3.74), the relations (2.3.70) and (2.3.71) are satisfied*
and, hence, all the results of Theorem 2.3.43 remain valid.

b. The finite element discretization

We consider again the operator*L*(with homogenous first boundary condition) in (2.3.65)
with sufficiently smooth coefficient functions. Then*L* can be written in the weak form as

follows Z
continuously differentiable w.r.t. *t* and belongs to *H*^{1}(Ω) for any fixed *t.*

In order to define a discrete finite element mesh operator, let*φ*_{1}*, . . . , φ**N*¯ be finite element
basis functions from *H*^{1}(Ω) with the property

*N*¯

X

*i=1*

*φ**i*(x)*≡*1 (2.3.75)

in ¯Ω. Applying these functions to the space discretization and the *θ-method to the time*
discretization, we arrive again at the discrete mesh operator in the canonical form (2.3.17).

Now the matrices M*∈*IR^{N×}^{N}^{¯} and K^{(n)}*∈*IR^{N×}^{N}^{¯}, respectively, have the elements
*M** _{i,j}* = (M

*)*

_{?}*R 1*

_{i,j}Ω*φ**i*(x) dx*,* *K*_{i,j}^{(n)} = (K_{?}^{(n)})* _{i,j}*R 1

Ω*φ**i*(x) dx*,* (2.3.76)
where M* _{?}* and K

^{(n)}

*are, respectively, the so-called mass and stiffness matrices with the entries*

_{?}Therefore, we can use Theorem 2.3.18.

For the row-sums of the matrix M, by using the relation (2.3.75), we get:

(Me)*i* =

*−*R 1
then additionally we assume that the finite element basis functions are non-negative, i.e.,
the condition

*φ** _{i}*(x)

*≥*0 (2.3.80)

is satisfied. Then R

Ω*−a*_{0}(x, t* _{n}*)φ

*(x) dx*

_{i}*≥*inf(−a

_{0})R

Ω*φ** _{i}*(x) dx. Hence, for this case
K

^{(n)}e

*≥*0. We can summarize our results as follows.

Theorem 2.3.44 *Let us assume that the finite element discrete mesh operator* *L, *
*de-fined by* (2.3.17)*and* (2.3.76)-(2.3.78)*for arbitrary finite element basis functions, is *
*non-negativity preserving. For* *a*0 = 0 *it has each of the DNP, DWMP, DSMP, DWBMP,*
*DSBMP and DMNC properties. When* *a*_{0} *is a non-positive, independent of* x, function,
*or, when it varies in* x*and non-positive and for the basis functions the condition* (2.3.80)
*is satisfied, then* *L* *has the DNP, DWMP, DWBMP and DMNC properties.*

In the sequel we deal with the problem of how the non-negativity of the discrete mesh operator can be guaranteed for the above cases.