ABSTRACT
The 2D hp technology discussed in the previous chapters applies to a large class of elliptic problems, including elliptic systems of equations. In this chapter, we present three examples of applications:
• A stationary heat conduction in an orthotropic material • Linear elasticity equations as an example of an elliptic system • A class of axisymmetric Maxwell problems that may be reduced to
the solution of a single elliptic equation in cylindrical coordinates (r, θ )
In this section, we deal with a more realistic problem∗ of heat conduction in nonhomogeneous materials. The governing equations are:
− ∂
∂xi
( ai j
∂u ∂xj
) = f in ,
ai j ∂u ∂xj
ni + βu = g on ,
where u denotes the unknown temperature field defined in domain (see Figure 15.1). Each colored (shaded) subdomain corresponds to a different material and conductivity ai j listed in the figure. All materials are orthotropic, i.e., the off-diagonal conductivities are zero but the conductivities in x1 and x2 directions may differ by several orders of magnitude. The geometry of the domain has been modeled with 4 × 8 = 32 rectangles (enumerated in the lexicographic order) shown in Figure 15.2 along with a (converged) numerical solution. The four sets of Cauchy boundary data, corresponding to the four edges of rectangular domain (marked with Roman numbers) are specified in
Figure 15.1. The anisotropy of the material causes strong internal boundary layers. Additionally, at every point in the domain at which three or more different materials meet, the derivatives of the solution are singular. This is a general phenomenon characterizing interface problems. Referring to 214 for details, we mention only that the singularities resulting from strong material discontinuities are stronger than those corresponding to reentrant corners. By selecting sufficiently “bad” material data, we can produce a solution u ∈ H1+ with arbitrary small , whereas the solution to the “crack problem” (compare the discussion of the L-shape domain problem in the previous section) “lives” in H3/2− , for any > 0. The complicated structure of the solution is reproduced by the optimal hp mesh corresponding to relative error of 1%, presented in Figure 15.3. Notice the anisotropic refinements, both in h and p, to capture the boundary layers. Resolution of the strong singularities resulting from the strong material contrasts resulted in up to 60 h refinements! A mesh with such a complexity cannot be produced “by hand.” Finally, Figure 15.4 presents the convergence history for both the hp algorithm and the h-adaptive algorithm
with quadratic elements. The h strategy initially wins in the preasymptotic range but eventually it slows down to an algebraic rate of convergence only, whereas the hp strategy delivers the expected exponential rates. Notice that the hp strategy starts winning at the error level of 15%! The desired error level of 1% can be reached with the h adaptivity and quadratic elements only for meshes with millions of d.o.f., and it is not accessible on a regular workstation. All routines corresponding to the battery problem can be found in the directory battery.
