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A matrix analysis approach to discrete comparison principles for nonmonotone PDE

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Abstract

We present a linear algebra approach to establishing a discrete comparison principle for a nonmonotone class of quasilinear elliptic partial differential equations. In the absence of a lower order term, local conditions on the mesh are required to establish the comparison principle and uniqueness of the piecewise linear finite element solution. We consider the assembled matrix corresponding to the linearized problem satisfied by the difference of two solutions to the nonlinear problem. Monotonicity of the assembled matrix establishes a maximum principle for the linear problem and a comparison principle for the nonlinear problem. The matrix analysis approach to the discrete comparison principle yields sharper constants and more relaxed mesh conditions than does the argument by contradiction used in previous work.

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Acknowledgements

SP would like to acknowldege Wright State University where a substantial portion of the writing was completed. Both authors would like to acknowledge the anonymous reviewers for suggestions that improved the clarity of the results.

Funding

SP was supported in part by NSF DMS 1719849 and 1852876. YZ was supported in part by NSF DMS 1319110.

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Correspondence to Sara Pollock.

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Appendix

Appendix

Here, we include the detailed arguments on the diagonal dominance requirements of matrix A, with A given by (3.4). In Lemma 4 we show A is GDD in accordance with Theorem 5. From the computations therein, we show in Lemma 3 that A is also DD+, in accordance with Theorem 4, with positive row sums for each index i such that vertex qi neighbors the Dirichlet boundary. Both GDD and DD+ properties hold on satisfaction of the conditions of Lemma 1.

We proceed to construct a positive diagonal matrix Dε for which all row sums of ADε are positive. The diagonal elements di of Dε are defined as an increasing sequence based on their distance from ΓD. First, we require a notion of distance from the boundary.

Definition 7

Let pi denote the length of the shortest path to a neighborhood of the Dirichlet boundary from vertex qi. In particular, if \(\bar {\mathcal Q}_{i} \setminus {\mathcal Q}_{i} \ne \emptyset \), then pi = 0. Otherwise, pi is defined to be the least number of edges traversed between qi and any vertex qj with pj = 0.

This notion of distance to the boundary is well-defined regardless of the number of connected components comprising domain Ω.

Lemma 4

Let Assumption1 and Assumption2 hold. LetA = (aij) be given by (3.4), and letA = (αij).Assume condition (4.1) of Lemma1 holds true, and forsome\(\bar \varepsilon > 0\)itholds as follows:

$$ \frac{K_{\eta} \beta_{M}\delta(u_{2})}{3\beta_{m}} < k_{\alpha} - \bar\varepsilon. $$
(6.1)

Let Dεbe the diagonal matrix with entries di given by the following:

$$ d_{i} = 1 - \varepsilon_{p_{i}}, \quad \varepsilon_{p_{i}} = \varepsilon_{p_{i}-1} - r^{p_{i}-1}\delta_{0}, ~p_{i} \ge 1, $$
(6.2)

for pigiven by Definition 7, fixed 0 < ε0 < 1, and 0 < r, δ0 < 1, to be defined below. Then, matrixADεis strictly diagonally dominant, and condition (6.1) relaxes to condition (4.1) for Ato be a Z-matrix, as\(\bar \varepsilon \rightarrow 0\).

Proof

First, it is noted that the sequence {εj} from (6.2) is a strictly decreasing sequence. By summing the geometric terms in (6.2), we also see the sequence {εj} is strictly positive if ε0 > δ0/(1 − r), which will be assured as \(\bar \varepsilon \rightarrow 0\) for fixed ε0. As a result, the coefficients di are ordered by the distance of each qi to the boundary according to Definition 7, and di → 1 + δ0/(1 − r) − ε0, for increasing pi.

By the positivity of the diagonals, and the nonpositivity of the off-diagonals, we require for each row i of the product ADε as follows:

$$ d_{i} \alpha_{ii} + \sum\limits_{j = 1, j \ne i}^{n} d_{j} \alpha_{ij} > 0. $$
(6.3)

By slight abuse of notation, let \(j \in {\mathcal Q}_{i}\) mean index j such that \(q_{j} \in {\mathcal Q}_{i}\). Let \(n = \text {card}({\mathcal Q})\), the number of mesh degrees of freedom. For conciseness, κ(x, u) will be denoted κ(u) in the remainder of the proof. Expanding (6.3) by (3.4) and rearranging terms yields the following:

$$ \begin{array}{@{}rcl@{}} d_{i} \alpha_{ii} + \!\!\sum\limits_{j = 1, j \ne i}^{n} \!d_{j} \alpha_{ij} & =& d_{i} {\int}_{\omega_{i}}\! \kappa(u_{1}) \nabla \varphi_{i} \!\cdot\! \nabla \varphi_{i} \mathrm{d}x + \sum\limits_{j \in {\mathcal Q}_{i}} d_{j} {\int}_{\omega_{ij}} \kappa(u_{1}) \nabla \varphi_{i} \!\cdot\! \nabla \varphi_{j} \mathrm{d}x \\ && + d_{i} {\int}_{\omega_{i}}\!\! \mathbf{b}(x,u_{2}) \!\cdot\! \nabla \varphi_{i} \varphi_{i} \mathrm{d}x + \sum\limits_{j \in {\mathcal Q}_{i}} d_{j} {\int}_{\omega_{ij}} \!\! \mathbf{b}(x,u_{2}) \!\cdot\! \nabla \varphi_{j} \varphi_{i} \mathrm{d}x \\ && + d_{i} {\int}_{\omega_{i}} c(x) {\varphi_{i}^{2}} \mathrm{d}x + \sum\limits_{j \in {\mathcal Q}_{i}} d_{j} {\int}_{\omega_{ij}} c(x) \varphi_{j} \varphi_{i} \mathrm{d}x. \end{array} $$
(6.4)

The contribution to the total sum from the last line of (6.4) is strictly nonnegative and need not be considered further. Each of the first two lines of (6.4) is now considered with respect to the membership of each \(q_{j} \in \bar {\mathcal Q}_{i}\) in one of three sets.

Define the sets as follows:

$$ {\mathcal Q}_{i}^{-1} = \bar {\mathcal Q}_{i} \setminus {\mathcal Q}_{i}, ~\text{ and }~ {{\mathcal Q}_{i}^{p}} := \{ q_{j} \in \bar {\mathcal Q}_{i} ~|~ p_{j} = p\}, ~p \ge 0. $$
(6.5)

A first key observation for the following analysis is for each vertex \(q_{j} \in \bar {\mathcal Q}_{i}\), qj is in exactly one of \({\mathcal Q}_{i}^{p_{i}-1}, {\mathcal Q}_{i}^{p_{i}}, {\mathcal Q}_{i}^{p_{i} + 1}\). A second key observation is at least one qj in \(\bar {\mathcal Q}_{i}\) is in \({\mathcal Q}_{i}^{p_{i}-1}\), meaning at least one neighbor of qi is closer in the sense of Definition 6.5 to ΓD. We now partition the terms of (6.4) into sums over each of these three sets. Again, let \(j \in {{\mathcal Q}_{i}^{p}}\) mean index j for which \(q_{j} \in {{\mathcal Q}_{i}^{p}}\), and for simplicity of notation, let p denote pi. If p = 0, meaning vertex qi neighbors the Dirichlet boundary, the contribution from the first line on the RHS of (6.4) is as follows:

$$ -d_{i} \sum\limits_{j \in {\mathcal Q}_{i}^{-1}} {\int}_{\omega_{ij}} \kappa(u_{1}) \nabla \varphi_{i} \!\cdot\!\nabla \varphi_{j} \mathrm{d}x + \sum\limits_{j\in {\mathcal Q}_{i}} (d_{j} - d_{i}){\int}_{\omega_{ij}} \kappa(u_{1}) \nabla \varphi_{i} \!\cdot\! \nabla \varphi_{j} \mathrm{d}x. $$
(6.6)

For p > 0, meaning vertex qi is without neighbors on ΓD, we have the following:

$$ \sum\limits_{j\in {\mathcal Q}_{i}} (d_{j} - d_{i}){\int}_{\omega_{ij}} \kappa(u_{1}) \nabla \varphi_{i} \!\cdot\! \nabla \varphi_{j} \mathrm{d}x. $$
(6.7)

For p = 0, the contribution from the second line of (6.4) can be written as follows:

$$ \begin{array}{@{}rcl@{}} && d_{i} \sum\limits_{j \in {\mathcal Q}_{i}^{-1}} (u_{2}(q_{j}) - u_{2}(q_{i})) {\int}_{\omega_{ij}} \nabla \varphi_{j} \!\cdot\! \nabla \varphi_{i} b(x) \varphi_{i} \mathrm{d}x \\ && + \sum\limits_{j \in {\mathcal Q}_{i}} (d_{j} - d_{i})(u_{2}(q_{i}) - u_{2}(q_{j})) {\int}_{\omega_{ij}} \nabla \varphi_{i} \!\cdot\! \nabla \varphi_{j} b(x) \varphi_{i} \mathrm{d}x \\ && + \sum\limits_{j \in {\mathcal Q}_{i}} d_{j} \sum\limits_{T \in \omega_{ij}} (u_{2}(q_{ij}^{T}) - u_{2}(q_{j})) {\int}_{T}\nabla \varphi_{ij}^{T} \!\cdot\! \nabla \varphi_{j} b(x) \varphi_{i} \mathrm{d}x. \end{array} $$
(6.8)

Let \({\mathcal K}_{ij}^{T} := (u_{2}(q_{ij}^{T}) - u_{2}(q_{j})) {\int }_{T}\nabla \varphi _{ij}^{T} \!\cdot \! \nabla \varphi _{j} b(x) \varphi _{i} \mathrm {d}x\). The third line of (6.8) can be expanded as follows:

$$ \begin{array}{@{}rcl@{}} && \left( d_{i}\sum\limits_{j \in \bar {\mathcal Q}_{i}}\sum\limits_{T \in \omega_{ij}} {\mathcal K}_{ij}^{T}\right) + \left( \sum\limits_{j \in {\mathcal Q}_{i}} (d_{j} - d_{i}) \sum\limits_{T \in \omega_{ij}} {\mathcal K}_{ij}^{T} \right) - \left( d_{i} \sum\limits_{j \in \bar {\mathcal Q}_{i} \setminus {\mathcal Q}_{i}} \sum\limits_{T \in \omega_{ij}} {\mathcal K}_{ij}^{T} \right) \\ && = \left( \sum\limits_{j \in {\mathcal Q}_{i}} (d_{j} - d_{i}) \sum\limits_{T \in \omega_{ij}} {\mathcal K}_{ij}^{T} \right) - \left( d_{i} \sum\limits_{j \in \bar {\mathcal Q}_{i} \setminus {\mathcal Q}_{i}} \sum\limits_{T \in \omega_{ij}} {\mathcal K}_{ij}^{T} \right), \end{array} $$
(6.9)

as the first term in the left of (6.9) is zero because the \({\mathcal K}_{ij}^{T}\) terms cancel pairwise when summed over the entire patch \(\bar {\mathcal Q}_{i}\).

For p > 0, the contribution from the second line of (6.4) can be written as follows:

$$ \sum\limits_{j \in {\mathcal Q}_{i}} (d_{j} - d_{i})(u_{2}(q_{i}) - u_{2}(q_{j})) {\int}_{\omega_{ij}} \nabla \varphi_{i} \!\cdot\! \nabla \varphi_{j} b(x) \varphi_{i} \mathrm{d}x + \sum\limits_{j \in {\mathcal Q}_{i}} d_{j}\sum\limits_{T \in \omega_{ij}} {\mathcal K}_{ij}^{T}, $$
(6.10)

where similarly to (6.9) but with \(\bar {\mathcal Q}_{i} \setminus {\mathcal Q}_{i} = \emptyset \), the last sum over \(j \in {\mathcal Q}_{i}\) can be written.

$$ \sum\limits_{j \in {\mathcal Q}_{i}} (d_{j} - d_{i}) \sum\limits_{T \in \omega_{ij}} {\mathcal K}_{ij}^{T}. $$
(6.11)

For the case p = 0, applying expansions (6.6), (6.8), and (6.9) to (6.4), we have the following:

$$ d_{i} \alpha_{ii} + \sum\limits_{j = 1, j \ne i} d_{j} \alpha_{ij} \ge -d_{i}\sum\limits_{j \in {\mathcal Q}_{i}^{-1}} {\mathcal J}_{ij} + \sum\limits_{j \in {\mathcal Q}_{i}} (d_{j} - d_{i}){\mathcal J}_{ij}, $$
(6.12)

with \({\mathcal J}_{ij}< 0\) given by the following:

$$ \begin{array}{@{}rcl@{}} {\mathcal J}_{ij} & :=& {\int}_{\omega_{ij}} (\kappa(x,u_{1}) + (u_{2}(q_{i}) - u_{2}(q_{j}))b(x) \varphi_{i})\nabla \varphi_{i} \!\cdot\! \nabla \varphi_{j} \mathrm{d}x + \sum\limits_{T \in \omega_{ij}} {\mathcal K}_{ij}^{T} \\ & \le& \frac 1 2 \left( -k_{\alpha} \beta_{m} + \frac{K_{\eta} \beta_{M}\delta_{\omega_{ij}}(u_{2})}{3} \right) < -\frac{\beta_{m}\bar \varepsilon}{2} < 0, \end{array} $$
(6.13)

where the first inequality follows by the angle conditions of Assumption 2 and the second by condition (6.1), as in (4.7)–(4.8) without the lower order term. It is also important to note that taking into consideration the upper bound on κ given in (2.1), we have finite numbers \(\mathcal {J}_{L}, {\mathcal {J}}^{U}\) for which as follows:

$$ \frac{\beta_{m} \bar \varepsilon}{2} \le {\mathcal J}_{L} \le |{\mathcal J}_{ij}| \le {\mathcal J}^{U}. $$
(6.14)

For p = 0, the sum over \({\mathcal Q}_{i}^{-1}\) contains at least one term and di = 1 − ε0, whereas djdi is either zero or ε0ε1. Let m be the maximum number of neighbors of any given vertex. A fixed maximum m is implied by the smallest-angle condition (2.6). Together with theses observations, applying (6.13) and (6.14) to (6.12) yields the following:

$$ d_{i} \alpha_{ii} + \sum\limits_{j = 1, j \ne i}^{n} d_{j} \alpha_{ij} \ge \frac{(1 - \varepsilon_{0})\beta_{m} \bar \varepsilon}{2} - (m-1) (\varepsilon_{0} - \varepsilon_{1}) {\mathcal J}^{u}. $$
(6.15)

From (6.2), ε0ε1 = δ0, so setting is as follows:

$$ \delta_{0} := \frac{\bar \varepsilon \beta_{m}(1 - \varepsilon_{0})}{2 m{\mathcal J}^{U}}, $$
(6.16)

forces the row sum in (6.15) to be strictly positive for the case p = 0.

For the case p > 0, applying expansions (6.7), (6.10), and (6.11) to (6.4), and \({\mathcal J}_{ij}< 0\) from (6.13), we have the following:

$$ \begin{array}{@{}rcl@{}} d_{i} \alpha_{ii} + \sum\limits_{j = 1, j \ne i} d_{j} \alpha_{ij} & =& \sum\limits_{j \in {\mathcal Q}_{i}^{p-1}} (d_{j} - d_{i}){\mathcal J}_{ij} + \sum\limits_{j \in {\mathcal Q}_{i}^{p+1}} (d_{j} - d_{i}){\mathcal J}_{ij} \\ & =& \sum\limits_{j \in {\mathcal Q}_{i}^{p-1}} (\varepsilon_{p} - \varepsilon_{p-1}){\mathcal J}_{ij} + \sum\limits_{j \in {\mathcal Q}_{i}^{p+1}} (\varepsilon_{p} - \varepsilon_{p+1}){\mathcal J}_{ij} \\ & \ge& (\varepsilon_{p-1} - \varepsilon_{p}) {\mathcal J}_{L} - (m-1) (\varepsilon_{p} - \varepsilon_{p+1}) {\mathcal J}^{U}, \end{array} $$
(6.17)

where the first sum must be nonempty because at least one vertex \(q_{j} \in {\mathcal Q}_{i}\) must be closer to the boundary than qi. From the construction (6.2), (6.17) then implies the following:

$$ d_{i} \alpha_{ii} + \sum\limits_{j = 1, j \ne i} d_{j} \alpha_{ij} \ge r^{p-1}\delta_{0} {\mathcal J}_{L} - (m-1)r^{p}\delta_{0} {\mathcal J}^{U}> 0, $$
(6.18)

for \(r < {\mathcal J}_{L}/((m-1) {\mathcal J}^{U})\). In particular, inequality (6.18) is satisfied for the following:

$$ r := \frac{ {\mathcal J}_{L}}{m {\mathcal J}^{U}} < 1. $$
(6.19)

It is finally noted for the sequence {εi} given by (6.2) as follows:

$$ \varepsilon_{i} > \varepsilon_{0} - \frac{\delta_{0}}{1-r} = \varepsilon_{0} \left( 1 + \frac{\bar \varepsilon \beta_{m}}{2(m{\mathcal J}^{U} - {\mathcal J}_{L})} \right) - \frac{\bar \varepsilon \beta_{m}}{2(m{\mathcal J}^{U} - {\mathcal J}_{L})} > 0, $$

for \(\bar \varepsilon \) small enough.

These arguments together show the matrix ADε is strictly diagonally dominant for diagonal matrix Dε defined by (6.2) with δ0 given by (6.16) and r given by (6.19). The remainder of the result follows by sending \(\bar \varepsilon \rightarrow 0\). □

Finally, we note that setting di = 1 for each i in the proof of Lemma 4 rather than rescaling the row sums as performed above provides a proof of Lemma 3 that A has the DD+ property required for Theorem 4, with positive row sums for each index i such that qi neighbors the boundary.

Proof (Lemma 3)

Indices i of A for which qi neighbors the Dirichlet boundary are seen to have positive row sums by setting di = dj = 1 in (6.12), and noting \({\mathcal J}_{ij} < 0\) under the given hypotheses. Indices i of A for which qi does not neighbor the boundary are seen to be zero by setting di = dj = 1 in the first line of (6.17). This establishes A is DD+ under the conditions that A (A) is a Z-matrix. □

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Pollock, S., Zhu, Y. A matrix analysis approach to discrete comparison principles for nonmonotone PDE. Numer Algor 83, 1007–1027 (2020). https://doi.org/10.1007/s11075-019-00713-x

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Keywords

  • Discrete comparison principle
  • Uniqueness
  • Nonmonotone problems
  • Quasilinear partial differential equations
  • Monotone matrix
  • M-matrix

Mathematics Subject Classification (2010)

  • 65N30
  • 35J62