Two-level discrete mesh operators

In the sequel, the values ν(x_i, n∆t) of the function ν defined in ¯Q_tM will be denoted by ν_iⁿ. Similar notation is applied to the function Lν. We introduce the vectors

νⁿ= [ν₁ⁿ, . . . , ν_Nⁿ_¯], νⁿ₀ = [ν₁ⁿ, . . . , ν_Nⁿ], νⁿ_∂ = [ν_N+1ⁿ , . . . , ν_Nⁿ_¯].

In many numerical methods, the discrete mesh operators have a special form, namely, they are defined as

(Lν)ⁿ_i = (X⁽ⁿ⁾₁ νⁿ−X⁽ⁿ⁾₂ νⁿ⁻¹)_i, i= 1, . . . , N, n = 1, . . . , M, (2.3.10) where X⁽ⁿ⁾₁ ,X⁽ⁿ⁾₂ ∈ IR^N×N¯ are some given matrices.³ In order to give the connections between the qualitative properties of such a type of mesh operators, we reformulate the conditions in Theorem 2.3.10, see Figure 2.3.1. We have already introduced the notation e= [1, . . . ,1]∈IR^N¯. TheN-element and the ( ¯N −N)-element version of this vector will be denoted by e₀ and e_∂, respectively, i.e., e= [e₀|e_∂]. Then the conditions L11 = 0 and L11≥0 read as

X⁽ⁿ⁾₁ e−X⁽ⁿ⁾₂ e= (X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e=0 and (X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e ≥0 (n = 1, . . . , M), respectively, while condition Ltt≥1 means that

X⁽ⁿ⁾₁ (∆tne)−X⁽ⁿ⁾₂ (∆t(n−1)e) = ∆t(n(X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e+X⁽ⁿ⁾₂ e)≥e₀.

If (X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e=0 (n= 1, . . . , M), then the above condition reduces to ∆tX⁽ⁿ⁾₂ e≥e₀. Hence, we have

3The term “two-level method” refers to the fact that two discrete time levels are involved into the definition of the mesh operator. Sometimes such a method is also called “one-step method”.

Theorem 2.3.13 If a non-negativity preserving discrete mesh operator of type (2.3.10) has such a structure that the conditions (X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e ≥ 0 and ∆t(n(X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e+ X⁽ⁿ⁾₂ e)≥e₀ hold, then the discrete weak maximum-minimum principles and the discrete maximum norm contractivity properties are always satisfied. If, in addition, (X⁽ⁿ⁾₁ − X⁽ⁿ⁾₂ )e = 0 and ∆tX⁽ⁿ⁾₂ e ≥ e0, then the operator possesses all the discrete qualitative properties introduced in Section 2.3.1.

As we can see from (2.3.10), the values (Lν)(xi, tn) (i= 1, . . . , N) depend only on the values of the function ν taken from the sets ¯P × {t_n}and ¯P × {t_n−1}. This suggests that the discrete qualitative properties can be written in such a form where only two levels in t are involved instead of all the levels from 0 to t^?. In order to define the qualitative properties in such a two-level form, we introduce the vector λⁿ₀ = [(Lν)ⁿ₁, . . . ,(Lν)ⁿ_N].

Let us consider the following property (denoted by DNP2) of a discrete mesh operator L: For any ν∈domL and n∈ {1, . . . , M} such that

νⁿ⁻¹₀ ,νⁿ⁻¹_∂ ,νⁿ_∂,λⁿ₀ ≥0, the relation νⁿ₀ ≥0 holds.

The two-level forms of the maximum-minimum principles can be formulated similarly as this is done in [46, 47, 57].

Theorem 2.3.14 For a discrete mesh operatorL in the form (2.3.10), the DNP property is equivalent to the DNP2 property.

Proof. First we prove that the DNP2 implies the DNP. Let ν ∈ domL and t^? = n^?∆t ∈ R_tM such that min_Gt? ν≥0 and Lν|_Q¯t? ≥0. Thus we have the relations

ν⁰₀,ν⁰_∂,ν¹_∂, . . . ,νⁿ_∂^?,λ¹₀,λ²₀, . . . ,λⁿ₀^? ≥0.

We need to show that ν|_Qt?¯ ≥ 0. Based on the property DNP2, ν⁰₀,ν⁰_∂,ν¹_∂,λ¹₀ ≥ 0 imply ν¹₀ ≥ 0. Similarly, ν¹₀,ν¹_∂,ν²_∂,λ²₀ ≥ 0 imply ν²₀ ≥ 0, and so on. At last, νⁿ₀^?−1,νⁿ_∂^?−1,νⁿ_∂^?,λⁿ₀^? ≥0 imply νⁿ₀^? ≥0. Thus, ν¹₀, . . . ,νⁿ₀^? ≥0 and ν|_Q¯t? ≥0.

Now let us suppose that the DNP property holds, and for a function ν ∈domL and n^? ∈ {1, . . . , M} the conditions νⁿ₀^?−1,νⁿ_∂^?−1,νⁿ_∂^?,λⁿ₀^? ≥ 0 are valid. Let us define the function ¯ν as follows: ¯ν_i⁰ = ν_i^n?−1, ¯ν_i¹ = ν_i^n? and ¯ν_iⁿ is chosen arbitrarily if 1 < n ≤ M (i = 1, . . . ,N¯). Applying the DNP property for the function ¯ν with t^? = ∆t, we obtain that ν¯¹₀ =νⁿ₀^? ≥0. This completes the proof.

The two-level form of the discrete non-negativity preservation property makes possible the formulation of its necessary and sufficient conditions. In order to give this condition in a linear algebraic form, we introduce the following convenient partitions of the matrices X⁽ⁿ⁾₁ and X⁽ⁿ⁾₂ :

X⁽ⁿ⁾₁ = [X⁽ⁿ⁾₁₀ |X⁽ⁿ⁾_1∂], X⁽ⁿ⁾₂ = [X⁽ⁿ⁾₂₀|X⁽ⁿ⁾_2∂],

where X⁽ⁿ⁾₁₀ and X⁽ⁿ⁾₂₀ are square matrices from IR^N×N, andX⁽ⁿ⁾_1∂,X⁽ⁿ⁾_2∂ ∈IR^N×N∂.

Theorem 2.3.15 Let us suppose that the matrices X⁽ⁿ⁾₁₀ (n = 1, . . . , M) of the discrete mesh operator L defined in (2.3.10) are regular. Then L possesses the discrete non-negativity preservation property (DNP or DNP2) if and only if the following relations hold for all n= 1, . . . , M,

(P1) (X⁽ⁿ⁾₁₀)⁻¹ ≥0, (P2) −(X⁽ⁿ⁾₁₀)⁻¹X⁽ⁿ⁾_1∂ ≥0, (P3) (X⁽ⁿ⁾₁₀)⁻¹X⁽ⁿ⁾₂ ≥0.

Proof. With the above notations and based on (2.3.10), we can write an identity in the following linear algebraic form

X⁽ⁿ⁾₁₀νⁿ₀ =−X⁽ⁿ⁾_1∂νⁿ_∂+X⁽ⁿ⁾₂ νⁿ⁻¹+λⁿ₀, (n = 1, . . . , M). (2.3.11) Supposing the regularity of the matrix X₁₀, we arrive at the iteration form

νⁿ₀ =−(X⁽ⁿ⁾₁₀)⁻¹X⁽ⁿ⁾_1∂νⁿ_∂+ (X⁽ⁿ⁾₁₀)⁻¹X⁽ⁿ⁾₂ νⁿ⁻¹+ (X⁽ⁿ⁾₁₀)⁻¹λⁿ₀, (n= 1, . . . , M).

According to the DNP2 property (which is equivalent to the DNP), the vectorνⁿ₀ is non-negative for all non-non-negative vectorsνⁿ⁻¹,νⁿ_∂ andλⁿ₀ if and only if the coefficient matrices (X⁽ⁿ⁾₁₀)⁻¹,−(X⁽ⁿ⁾₁₀ )⁻¹X⁽ⁿ⁾_1∂ and (X⁽ⁿ⁾₁₀)⁻¹X⁽ⁿ⁾₂ are non-negative matrices. This completes the proof.

Summarizing the results of the above three theorems, we can conclude the following.

Theorem 2.3.16 Let us assume that the non-negativity assumption (conditions (P1)-(P3)) is satisfied. Then, besides the DNP property,

• under the conditions

(X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e≥0; and ∆t(n(X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e+X⁽ⁿ⁾₂ e)≥e0 (2.3.12) the qualitative properties DWMP, DWBMP and DMNC;

• under the conditions

(X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e=0 and ∆tX⁽ⁿ⁾₂ e≥e₀ (or ∆tX⁽ⁿ⁾₁ e≥e₀) (2.3.13) the qualitative properties DWMP, DSMP, DWBMP, DSBMP and DMNC

are valid.

According to Theorem 2.3.12, for the mesh functions from H0 and H1 the conditions can be relaxed, namely, the second condition in (2.3.12) and (2.3.13) can be neglected.

Hence, we get

Theorem 2.3.17 Let us assume that the non-negativity assumption (conditions (P1)-(P3)) is satisfied. Then, besides the DNP property,

• under the conditions

(X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e≥0 (2.3.14) the qualitative properties DWMP, DWBMP and DMNC;

• under the conditions

(X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ )e=0 (2.3.15) the qualitative properties DWMP, DSMP, DWBMP, DSBMP and DMNC

are also valid for any mesh function from H₀ and H₁.

The operator L on the n-th time level is completely defined by the matrices X⁽ⁿ⁾₁ and X⁽ⁿ⁾₂ , see (2.3.10). In the typical numerical applications, as also in our work, (cf. Section 2.3.5), they are derived from the approximation of the concrete continuous operator L.

When we use a one-parameter family of the approximation (which is called theθ-method), these matrices are defined by the matricesM⁽ⁿ⁾,K⁽ⁿ⁾, (called mass and stiffness matrices, respectively), and a real parameter θ, as follows

X⁽ⁿ⁾₁ = 1

∆tM⁽ⁿ⁾+θK⁽ⁿ⁾, X⁽ⁿ⁾₂ = 1

∆tM⁽ⁿ⁾−(1−θ)K⁽ⁿ⁾.

(2.3.16)

The matrices M⁽ⁿ⁾ and K⁽ⁿ⁾ have the size N ×N¯. Hence, the discrete mesh operator L in (2.3.10) can be written in the following (so-called canonical) form:

(Lν)ⁿ_i = (M⁽ⁿ⁾νⁿ−νⁿ⁻¹

∆t +θK⁽ⁿ⁾νⁿ+ (1−θ)K⁽ⁿ⁾νⁿ⁻¹)_i. (2.3.17) Therefore, conditions (2.3.12) and (2.3.13) in Theorem 2.3.16 can be formulated as follows:

K⁽ⁿ⁾e≥0 and ∆t(n−1 +θ)K⁽ⁿ⁾e+M⁽ⁿ⁾e≥e₀; (2.3.12⁰) and

K⁽ⁿ⁾e =0 and M⁽ⁿ⁾e≥e₀. (2.3.13⁰) Thus, we have

Theorem 2.3.18 Let us assume that the discrete mesh operator L, defined by (2.3.10) and (2.3.16), is non-negativity preserving and the relation

M⁽ⁿ⁾e≥e0 (2.3.18)

holds. Then, beyond the DNP property, under the condition K⁽ⁿ⁾e ≥0, the operator L is DWMP, DWBMP and DMNC; while in the case K⁽ⁿ⁾e =0 it obeys each of the DWMP, DSMP, DWBMP, DSBMP and DMNC properties.

Remark 2.3.19 Let us note that, according to Theorem 2.3.12, the assumption is simpler for the mesh functions from H₁: if L is a discrete non-negativity preserving operator and the condition

K⁽ⁿ⁾e≥0 (2.3.19)

is satisfied, then it possesses all the other qualitative properties, too.

Remark 2.3.20 The above consideration is a special case of the following general ap-proach. For the pair of the matrices (X⁽ⁿ⁾₁ , X⁽ⁿ⁾₂ ) (which define the two-level operator) we define a mapping ϕ : IR^N×N¯ ×IR^N×N¯ → IR^N×N¯ ×IR^N×N¯ which is assumed to be a bijection. Then, we write the conditions, obtained for the matrices X⁽ⁿ⁾₁ and X⁽ⁿ⁾₂ , for the matrices ϕ⁻¹(X⁽ⁿ⁾₁ , X⁽ⁿ⁾₂ ), denoted by (M⁽ⁿ⁾, K⁽ⁿ⁾).

For the θ-method, as one can see from (2.3.16), the inverse of this bijection is ϕ⁻¹_θ (A,B) :=

µ 1

∆tA+θB, 1

∆tA−(1−θ)B

¶

, (2.3.20)

where θ is any fixed number. Hence,

ϕ_θ(A,B) = (∆t((1−θ)A+θB),A−B). (2.3.21) This means the following. The above approach (knowing M⁽ⁿ⁾ and K⁽ⁿ⁾ and selecting the mapping, i.e., by fixingθ, we define X⁽ⁿ⁾₁ andX⁽ⁿ⁾₂ ) can be reversed: when the matrices X⁽ⁿ⁾₁ and X⁽ⁿ⁾₂ are a priori given, and we know that the discretization was obtained by use of the θ-method, we proceed as follows. We introduce the matrices

M⁽ⁿ⁾ = ∆t

³

(1−θ)X⁽ⁿ⁾₁ +θX⁽ⁿ⁾₂

´

, K⁽ⁿ⁾=X⁽ⁿ⁾₁ −X⁽ⁿ⁾₂ , (2.3.22) and use the same conditions. However, as we will see later, the first approach is more natural, because, as it was already mentioned, we define the matrices M⁽ⁿ⁾ and K⁽ⁿ⁾ a priori from the approximation of the continuous operator.

In document Numerical Treatment of Linear Parabolic Problems (Pldal 24-28)