"... In PAGE 3: ... THE GOVERNING EQUATIONS AND THE NUMERICAL FORMULATION To illustrate the usefulness of the multilevel approach we briefly present a comparison of the one- and two-level DD preconditioners applied to the solution of two thermal convection problems and a Stokes flow problem. The governing elliptic PDEs (Navier-Stokes with the thermal energy equations) are shown in Table1 . These equations are discretized by a Galerkin least squares (GLS) finite element formulation which allows for equal order of interpolation of velocity and pressure[8].... In PAGE 3: ... In this standard 2D benchmark problem [11], a thermal con- vection (or buoyancy-driven) flow in a differentially heated square box in the presence of gravity is modeled. The momentum transport, energy transport and total mass conservation equations defined in Table1 are solved on a unit square. No-slip boundary conditions are applied on all walls.... ..."

"... In PAGE 13: ...Table1 shows a summary of the various features of the PDES languages outlined above. Note that the Statistics feature denotes whether the PDES language has built-in functions for statistical purposes.... ..."

"... In PAGE 13: ...Table1 shows a summary of the various features of the PDES languages outlined above. Note that the Statistics feature denotes whether the PDES language has built-in functions for statistical purposes.... ..."

"... In PAGE 12: ... 4 Field theory and multisymplectic geometry 4.1 Geometric formulation of field theory As one may see in Table1 , many of the main pdes in Physics can be obtained from a Lagrangian function. There are other interesting pdes describing dynamical processes, which are also variational with other conditions: constrained problems, higher-order Lagrangians, etc.... ..."

"... In PAGE 13: ... Table2 Selected Performance Measurement for Different PDES Languages As can be seen from Table 2, all packages reported positive speedup. The initial performance figure of APOSTLE, as indicated in [11], is speedup of 2.... In PAGE 13: ...Table 2 Selected Performance Measurement for Different PDES Languages As can be seen from Table2 , all packages reported positive speedup. The initial performance figure of APOSTLE, as indicated in [11], is speedup of 2.... ..."

"... In PAGE 13: ... Table2 Selected Performance Measurement for Different PDES Languages As can be seen from Table 2, all packages reported positive speedup. The initial performance figure of APOSTLE, as indicated in [11], is speedup of 2.... In PAGE 13: ...Table 2 Selected Performance Measurement for Different PDES Languages As can be seen from Table2 , all packages reported positive speedup. The initial performance figure of APOSTLE, as indicated in [11], is speedup of 2.... ..."

"... In PAGE 14: ...rdinary di erential equation with respect to x. Solving it yields formulae 6 from Table 2. The other cases are dealt with in an analogous way. Now we proceed to calculating the initial conditions (12) such that the initial value prob- lems for conditionally invariant PDEs (27) given in Table2 are reducible to Cauchy problems... In PAGE 15: ...of all we note that for equations 2, 3 of Table2 , the condition b of Theorem 3 fails to hold. For this reason, the corresponding initial value problem is not reduced to a Cauchy problem within the framework of our approach.... ..."