In Part II we shall work with parabolic partial differential equations on the rectangle (0, 1) × (0, T] in the space-time domain, where T is some fixed positive time. It is not essential to have a rectangle; one can transform many other domains to rectangular form.

2. Analytical Behaviour of Solutions

For a general discussion of the properties enjoyed by solutions of parabolic differential equations, the standard reference books are Friedman [Fri64] and Ladyženskaja et al. [LSU67]. The broad analysis presented there is classical in nature, dealing with solutions lying in Hölder spaces. Here we shall state only those fundamental results from [Fri64] that are necessary to provide a basis for our later work.

3. Finite Difference Methods

4. Finite Element Methods

From Part I we know that standard Galerkin finite element methods on equidistant meshes yield inaccurate approximate solutions of singularly perturbed two-point boundary value problems unless a large number of mesh points are used. The same disappointing behaviour occurs arises when dealing with parabolic convection-diffusion problems, because such methods have no built-in upwinding. Finite element methods will now be developed specifically for the convection-diffusion situation, either by choosing special basis functions or by working on meshes designed for these problems.

5. Two Adaptive Methods

All methods considered so far in Part II use fixed meshes that are chosen a priori. Adaptive methods, which were applied to two-point boundary value problems in Section I.2.5, aim to produce accurate numerical solutions by refining the mesh in certain regions, using the current computed solution to (4.1) as a guide to this refinement. In the present Chapter we consider two such methods.

Elliptic and Parabolic Problems in Several Space Dimensions

Frontmatter

1. Analytical Behaviour of Solutions

Notation: throughout Chapter 1, a generic point in Ω ⊂ ℝ^d is denoted by x = (x₁, x₂,…, x_d), but in the case d = 2 the notation (x, y) is sometimes used instead. The standard Sobolev spaces H^k(Ω) with associated norms ∥ · ∥_k and seminorms ¦·¦_k are often used, as their more general counterparts W^k,p(Ω) with seminorm ¦ · ¦_w^k,p(Ω).

2. Finite Difference Methods

3. Finite Element Methods

4. Time-Dependent Problems

In this chapter we discuss convection-diffusion and reaction-diffusion problems whose solutions are time-dependent (as in Part II) and are functions of more than one space variable (as in the preceding chapters of Part III). The analysis and numerical methods used are often combinations and extensions of techniques that appeared in this earlier material, so the chapter is relatively short, but new ideas such as dimension-splitting will also make their debuts.

The Incompressible Navier-Stokes Equations

Frontmatter

1. Existence and Uniqueness Results

2. Upwind Finite Element Method

3. Higher-Order Methods of Streamline Diffusion Type

4. Local Projection Stabilization for Equal-Order Interpolation

Local projection stabilization (LPS) was introduced in Part III, Chapter 3 for a scalar convection-diffusion equation. It will now be extended to the Oseen system

$$-\nu \Delta {\bf u} + ({\bf b} \cdot \nabla){\bf u} + \sigma {\bf u} + \nabla p = {\bf f} \; {\rm in} \; \Omega \subset {{\bf R}^d}$$

(4.1a)

$$\nabla \cdot {\bf u} = 0 \; {\rm in} \; \Omega$$

(4.1b)

$${\bf u} = {\bf 0} \; {\rm on} \; \partial\Omega$$

(4.1c)

As we saw in Chapter 3, the streamline diffusion method (SDFEM) can handle two types of instabilities: that caused by a violation of the discrete infsup (Babu?ska-Brezzi) condition (2.1) and that due to dominant convection. The SDFEM combines the Pressure Stabilized Petrov-Galerkin (PSPG) approach (testing the residual against ${\nabla{\rm q}}$) with the Streamline Upwind Petrov- Galerkin (SUPG) technique (testing the residual against ${({\rm b}\cdot\nabla)}{\bf v}$)

Despite the extensive theoretical and practical development of the SDFEM, a fundamental flaw in the method - in particular for higher-order interpolations - is that various terms must be added to the weak formulation to guarantee its consistency. Moreover, the requirement of consistency leads to undesirable effects when using residual-based stabilization methods like the SDFEM in optimal control problems; see the discussion at the beginning of Section III.3.3. LPS relaxes the consistency requirement while preserving the main features of the SDFEM approach; in particular, one can use equal-order interpolation without worrying about the Babuška-Brezzi condition. Furthermore, LPS allows us to separate velocity and pressure in the stabilization terms, which for systems of equations means that one can avoid non-physical couplings.

5. Local Projection Method for Inf-Sup Stable Elements

The previous chapter showed that local projection stabilization (LPS) for equal-order interpolation can handle two types of instabilities — that caused by a violation of the discrete inf-sup condition and that due to dominant convection in the case of high Reynolds number. But the flow problem is often only part of a coupled flow-transport problem; in the next chapter we shall see that mass conservation in the transport equation depends on the properties of the discrete velocity and in particular on the satisfaction of the incompressibility constraint. Unfortunately, when LPS is applied with equalorder interpolation, the discrete divergence-free property of the velocity field is disturbed by the term

$$\sum \limits_{M\in{\cal M}_h} \alpha_M(\kappa_h\nabla{p_h}, \kappa_h \nabla{q_h}) M$$

that stabilizes the pressure. For inf-sup stable finite element pairs, this pressure stabilization is unnecessary and we are faced only with the instability caused by dominant convection. Thus it is of interest to consider local projection stabilization for inf-sup stable finite elements.

The main objective of this chapter is an analysis of convergence properties of LPS applied to inf-sup stable discretizations of the Oseen problem.We shall restrict our attention to the enrichment variant of LPS and to a stabilizing term that controls separately fluctuations of the derivative in the streamline direction and fluctuations of the divergence.

6. Mass Conservation for Coupled Flow-Transport Problems

In this chapter we examine mass-conservation properties of finite element discretizations of coupled flow-transport problems. The system under consideration is described by the unsteady incompressible Navier-Stokes equations and a time-dependent transport equation; see [GS00a, GS00b, Hir88, Hir90] for models where this combination arises. The incompressibility constraint implies that global mass is conserved in the weak solution of the transport equation. Since the discretized velocity only satisfies a discrete incompressibility constraint, global mass is in general conserved only approximately in the numerical scheme. We shall investigate conditions under which discrete global mass conservation can be guaranteed.

7. Adaptive Error Control

The derivation of reliable and efficient a posteriori error estimates for the Navier-Stokes equations is an important consideration in computational fluid dynamics. On perusing existing a priori error estimates for the Navier-Stokes equations, one notices fundamental differences from and fresh difficulties compared with the estimates for diffusion-dominated and convection-dominated elliptic problems in Chapter II.3.6. Some new obstacles that appear are

a smallness condition on the Reynolds number Re = 1/v to ensure uniqueness of the solution (Chapter 1)
a well-posedness assumption for a linearized problem with an a priori error estimate that depends strongly on an unknown stability constant (Theorem 3.14).

As we saw in Example 3.16, this stability constant can grow exponentially in the Reynolds number Re. One should be aware that such a property will restrict considerably the quantitative value of our estimates. On the other hand, in the special case of a no-flow problem (see Remark 2.10), one has uniqueness of the solution for all Reynolds numbers and error estimates with a right-hand side that is a polynomial function of Re.

Backmatter

Titel: Robust Numerical Methods for Singularly Perturbed Differential Equations
verfasst von: Hans-Görg Roos
Martin Stynes
Lutz Tobiska
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-34467-4
Print ISBN: 978-3-540-34466-7
DOI: https://doi.org/10.1007/978-3-540-34467-4