For ease of reading, we start with a brief outline of the main content.
3.3 Well Posed Problems and Stability
In this section, we define the concepts needed in the rest of the paper, and most of the material can be found in [
2,
21,
22]. Roughly speaking, an initial boundary value problem is well posed if a unique solution that depends continuously on the initial and boundary data exists. Consider the following general linear initial boundary value problem
$$\begin{aligned} W_t+\mathscr {P}W= & {} \mathbf {F},\quad \mathbf {x} \in {\varOmega }, \quad t\ge 0 \nonumber \\ \mathscr {L}W= & {} \mathbf {g}, \quad \mathbf {x} \in \partial {\varOmega }, \quad t\ge 0 \nonumber \\ W= & {} \mathbf {f},\quad \mathbf {x} \in {\varOmega }, \quad t= 0 \end{aligned}$$
(3.8)
where
W is the solution,
\(\mathscr {P}\) is the spatial differential operator and
\(\mathscr {L}\) is the boundary operator. In this paper,
\(\mathscr {P}\) and
\(\mathscr {L}\) are linear operators,
\(\mathbf {F}\) is a forcing function, and
\(\mathbf {g}\) and
\(\mathbf {f}\) are boundary and initial functions, respectively.
\(\mathbf {F}\),
\(\mathbf {g}\) and
\(\mathbf {f}\) are the known data of the problem. In this paper we consider smooth and compatible data leading to sufficiently smooth solutions. The initial boundary value problem (
3.8) is posed on the domain
\( {\varOmega }\) with boundary
\(\partial {\varOmega }\).
We introduce the scalar product and norm as
$$\begin{aligned} (U,V)_{{\varOmega }}= \displaystyle \int _{{\varOmega }} U^T H V \, dx \, dy \, dz, \quad \Vert U(\cdot ,t)\Vert ^2_{{\varOmega }}=(U,U)_{{\varOmega }}, \end{aligned}$$
(3.9)
for real valued vector functions
U,
V and a positive definite symmetric matrix
H.
If a solution to (
3.8) exist, semi-boundedness of
\(\mathscr {P}\) leads directly to well-posedness. However, with too many boundary conditions, existence is not guaranteed. Consequently, a more restrictive definition is required.
The energy method (which we will describe in detail in Sect.
4.1 below) and maximally semi-bounded operators lead directly to well-posed problems.
For certain classes of problems with specific types of boundary conditions, the energy method in combination with maximally semi-boundedness operators lead to even stronger estimates, and so called strongly well-posed problems.
Closely related to well-posedness is the concept of stability. The semi-discrete version of (
3.8) is
$$\begin{aligned} (W_j)_t+{\mathscr {Q}} W_j= & {} \mathbf {F}_j,\quad \mathbf {x}_j \in {\varOmega }, \quad t\ge 0 \nonumber \\ {\mathscr {M}} W_j= & {} \mathbf {g}_j, \quad \mathbf {x}_j \in \partial {\varOmega }, \quad t\ge 0 \nonumber \\ W_j= & {} \mathbf {f}_j,\quad \mathbf {x}_j \in {\varOmega }, \quad t= 0. \end{aligned}$$
(3.13)
The difference operator
\(\mathscr {Q}\) approximates the differential operator
\(\mathscr {P}\) and the discrete boundary operator
\(\mathscr {M}\) approximates
\(\mathscr {L}\).
\(\mathbf {F}_j\),
\(\mathbf {g}_j\) and
\(\mathbf {f}_j\) are the known smooth compatible data of the problem (
3.8) injected on the grid
\( \mathbf {x}_j=(x_j,y_j,z_j)\). The difference approximation (
3.13) is a consistent approximation of (
3.8).
We now define semi-bounded discrete operators in analogy with differential operators. Let the volume element corresponding to the
\(j\mathrm{th}\) node be
\({\varDelta }{\varOmega }_j\). The discrete scalar product and norm are defined by
$$\begin{aligned} (U,V)_{{\varOmega }_h}= \displaystyle \sum _{j=1}^{j=N} U^T_j H_j V_j {\varDelta }{\varOmega }_j, \quad \Vert U(\cdot ,t)\Vert ^2_{{\varOmega }_h}=(U,U)_{{\varOmega }_h}, \end{aligned}$$
(3.14)
for real valued vector functions
\(U_j,V_j\) and positive definite symmetric matrices
\(H_j\).
Unlike in the continuous case, the problem with existence and uniqueness related to the number of boundary conditions does not exist in the discrete case. The number of boundary conditions (including numerical ones) is simply equal to the number of linearly independent conditions in \(\mathscr {M} W_j=\mathbf {g}_j\) that are required for the semi-discrete system to have a unique solution. Different numerical boundary conditions can lead to different solutions on coarse grids. However, for sufficiently fine meshes and stable approximations, the numerical solution will converge to the continuous unique solution. Hence we need not restrict semi-boundedness to maximal semi-boundedness as was done for the continuous case above.
The discrete energy method (which we will describe in detail in Sect.
5.2 below) and semi-bounded operators lead directly to stability.
As in the continuous case, for certain classes of problems with specific types of boundary conditions, the energy method in combination with semi-bounded operators can lead to even stronger estimates, and so called strongly stable problems.
The definitions of well-posedness and stability above are strikingly similar. However, the bounds in the corresponding estimates need not be the same. The following definition connects the growth rates of the continuous and semi-discrete solutions.