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Remark 1. Upon appropriate simplification, error representation formulas for a system comprising two coupled elliptic problems and for a system comprising a parabolic problem coupled to an elliptic problem, follow immediately from the above result. Using the initial conditions uly = 7Tui,9 and ¢;,n = 0 to rearrange the first two § O+ g' terms we obtain the representation formula (3.6), We now consider corrections to the error representation formula due to errors or the boundary and error in the adjoint solution.

Figure 1 Remark 1. Upon appropriate simplification, error representation formulas for a system comprising two coupled elliptic problems and for a system comprising a parabolic problem coupled to an elliptic problem, follow immediately from the above result. Using the initial conditions uly = 7Tui,9 and ¢;,n = 0 to rearrange the first two § O+ g' terms we obtain the representation formula (3.6), We now consider corrections to the error representation formula due to errors or the boundary and error in the adjoint solution.

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We begin by examining the “domain decomposition” scenario in which there is a single operator and a single solution within the entire domain which we choose to subdivide for computational purposes. We then consider problems in which distinct operators and/or distinct solutions pertain to the two subdomains, including the case in which an elliptic operator in one domain is coupled to a parabolic operator in the other domain. In this case the arguments described in section 3 must be suitably (though easily) modified. We also consider situations in which the discretizations do not conform along the interface, the situation arising in mortar finite element methods. In all cases, (3.10) provides an accurate estimate of the numerical error which improves as the adjoint solution is refined.
We begin by examining the “domain decomposition” scenario in which there is a single operator and a single solution within the entire domain which we choose to subdivide for computational purposes. We then consider problems in which distinct operators and/or distinct solutions pertain to the two subdomains, including the case in which an elliptic operator in one domain is coupled to a parabolic operator in the other domain. In this case the arguments described in section 3 must be suitably (though easily) modified. We also consider situations in which the discretizations do not conform along the interface, the situation arising in mortar finite element methods. In all cases, (3.10) provides an accurate estimate of the numerical error which improves as the adjoint solution is refined.
Subsection 5.1. Effectivity ratios R and R, for model problems (i)—(v) varying the spatial accu racy of the adjoint solution. The value of r corresponds to the number of uniform spatial refinement used to obtain the adjoint mesh from the adapted mesh on which the solution is computed. The prob lem was solved with Nz = 5 time steps, final time T = 1, and the original mesh of DOF = 78 degree of freedom.
Subsection 5.1. Effectivity ratios R and R, for model problems (i)—(v) varying the spatial accu racy of the adjoint solution. The value of r corresponds to the number of uniform spatial refinement used to obtain the adjoint mesh from the adapted mesh on which the solution is computed. The prob lem was solved with Nz = 5 time steps, final time T = 1, and the original mesh of DOF = 78 degree of freedom.
Subsection 5.1.2. Effectivity ratios R and R, varying the spatial accuracy of the adjoint solution. The value of r corresponds to the number of uniform spatial refinements used to obtain the adjoint mesh from the adapted mesh on which the solution is computed. The problem was solved with N; = 5 time steps, final time T = 1, and the original mesh of DOF = 253 degrees of freedom for the conforming (conf) meshes and DOF = 393 for the nonconforming (nonconf) meshes. In order to calculate purely interfacial error, we set v, = wo =O andy =T-t (to satisfy the condition at t = T). We report effectivity ratios for the interfacial error in columns four and five of Table 5.2.
Subsection 5.1.2. Effectivity ratios R and R, varying the spatial accuracy of the adjoint solution. The value of r corresponds to the number of uniform spatial refinements used to obtain the adjoint mesh from the adapted mesh on which the solution is computed. The problem was solved with N; = 5 time steps, final time T = 1, and the original mesh of DOF = 253 degrees of freedom for the conforming (conf) meshes and DOF = 393 for the nonconforming (nonconf) meshes. In order to calculate purely interfacial error, we set v, = wo =O andy =T-t (to satisfy the condition at t = T). We report effectivity ratios for the interfacial error in columns four and five of Table 5.2.
Fic. 5.1. Conforming versus nonconforming meshes in domains Q, and Q2 with a common interface T. We consider both triangulations in Q; and Q2 that conform on the interface (see Figure 5.1(a)) and those that do not (see Figure 5.1(b)). Conforming discretizations have 121 degrees of freedom in each domain and 11 on the interface. Nonconforming discretizations have 121 degrees of freedom in 21, 256 in Q2, and 16 on T. As seen in Table 5.2, the calculated effectivity ratios R for both conforming and nonconforming cases converge to one as the adjoint mesh is refined.
Fic. 5.1. Conforming versus nonconforming meshes in domains Q, and Q2 with a common interface T. We consider both triangulations in Q; and Q2 that conform on the interface (see Figure 5.1(a)) and those that do not (see Figure 5.1(b)). Conforming discretizations have 121 degrees of freedom in each domain and 11 on the interface. Nonconforming discretizations have 121 degrees of freedom in 21, 256 in Q2, and 16 on T. As seen in Table 5.2, the calculated effectivity ratios R for both conforming and nonconforming cases converge to one as the adjoint mesh is refined.
Subsection 5.2. Effectivity ratios R and R, for model problems (vii)-(ix) varying the spatia accuracy of the adjoint solution. The value of r corresponds to the number of uniform spatial refine- ments used to obtain the adjoint mesh from the adapted mesh on which the solution is computed. The problem was solved with Ny = 5 time steps, final time T = 1, and the original mesh of DOF = 78 de- grees of freedom for the conforming (conf) meshes and DOF = 108 for the nonconforming (nonconf, meshes. Once again summing the absolute values of the error contributions on each element and eliminating the possibility for cancelation of error between elements, we compute the effectivity ratios shown in Table 5.3.
Subsection 5.2. Effectivity ratios R and R, for model problems (vii)-(ix) varying the spatia accuracy of the adjoint solution. The value of r corresponds to the number of uniform spatial refine- ments used to obtain the adjoint mesh from the adapted mesh on which the solution is computed. The problem was solved with Ny = 5 time steps, final time T = 1, and the original mesh of DOF = 78 de- grees of freedom for the conforming (conf) meshes and DOF = 108 for the nonconforming (nonconf, meshes. Once again summing the absolute values of the error contributions on each element and eliminating the possibility for cancelation of error between elements, we compute the effectivity ratios shown in Table 5.3.
Fic. 6.1. Subsection 6.1. Meshes after space refinement using quadratic finite-elements in Qy and linear finite-elements in Q2. (a) Refinement based on error in the domain. (b) Refinement based on error on the interface. The computed errors, shown in Table 6.1, decrease as the mesh is refined, although not necessarily as dramatically as the errors in the quantity of interest.
Fic. 6.1. Subsection 6.1. Meshes after space refinement using quadratic finite-elements in Qy and linear finite-elements in Q2. (a) Refinement based on error in the domain. (b) Refinement based on error on the interface. The computed errors, shown in Table 6.1, decrease as the mesh is refined, although not necessarily as dramatically as the errors in the quantity of interest.
Subsection 6.1. Estimated total errors and effectivity ratios for both original and refined meshes using quadratic finite-elements in Q1 and linear finite-elements in Q2. DOF, and DOF? are the numbers of degrees of freedom in each sub-domain. TABLE 6.1
Subsection 6.1. Estimated total errors and effectivity ratios for both original and refined meshes using quadratic finite-elements in Q1 and linear finite-elements in Q2. DOF, and DOF? are the numbers of degrees of freedom in each sub-domain. TABLE 6.1
FIG. 6.2. Subsections 6.2 and 6.3. Cartoon of the solutions as a function of time This solution is a Gaussian with a peak located at (0.5, 1.5) at time t = 0, and moving clockwise along an elliptic trajectory inside 0; UQ2. At the final time T = 2, the peak returns to its initial position after making one complete orbit. A snapshot view of the solution is presented in Figure 6.2(a). Unless stated otherwise, we began the adaptive procedure with a uniform mesh containing 453 degrees of freedom, and 200 equally distributed time steps. 6.2.1. Adaptivity based on error in the domain only. First, we consider che quantity of interest representing the average in the whole domain, so that 6.2.1. Adaptivity based on error in the domain only. First, we consider
FIG. 6.2. Subsections 6.2 and 6.3. Cartoon of the solutions as a function of time This solution is a Gaussian with a peak located at (0.5, 1.5) at time t = 0, and moving clockwise along an elliptic trajectory inside 0; UQ2. At the final time T = 2, the peak returns to its initial position after making one complete orbit. A snapshot view of the solution is presented in Figure 6.2(a). Unless stated otherwise, we began the adaptive procedure with a uniform mesh containing 453 degrees of freedom, and 200 equally distributed time steps. 6.2.1. Adaptivity based on error in the domain only. First, we consider che quantity of interest representing the average in the whole domain, so that 6.2.1. Adaptivity based on error in the domain only. First, we consider
FIG. 6.3. Subsection 6.2.1. Space/time blocks and temporal refinements produced for the quan tity of interest equal to the average solution over the whole domain Qy UQe2 given NMAX = 400. The time-space adaptive refinement strategy applied to the original uniform mesl produced six temporal blocks ,,..., Jg shown in Figure 6.3. As expected, the spatia mesh refinements trace the movement of the Gaussian shaped “peak” of the solutior in the domain. Away from the peak of the Gaussian, where the solution approache: a constant, the original mesh provides sufficiently good resolution and need not be refined.
FIG. 6.3. Subsection 6.2.1. Space/time blocks and temporal refinements produced for the quan tity of interest equal to the average solution over the whole domain Qy UQe2 given NMAX = 400. The time-space adaptive refinement strategy applied to the original uniform mesl produced six temporal blocks ,,..., Jg shown in Figure 6.3. As expected, the spatia mesh refinements trace the movement of the Gaussian shaped “peak” of the solutior in the domain. Away from the peak of the Gaussian, where the solution approache: a constant, the original mesh provides sufficiently good resolution and need not be refined.
Subsection 6.2.1. Estimated total errors Eest and Eest for the quantity of interest equal to the average solution over the whole domain, the temporal and spatial components of the error Ez anc Ex, and the corresponding L2-error eg calculated on the original uniform mesh and on the refines mesh with NUAX = 400. DOF is the maximum number of degrees of freedom in a temporal block
Subsection 6.2.1. Estimated total errors Eest and Eest for the quantity of interest equal to the average solution over the whole domain, the temporal and spatial components of the error Ez anc Ex, and the corresponding L2-error eg calculated on the original uniform mesh and on the refines mesh with NUAX = 400. DOF is the maximum number of degrees of freedom in a temporal block
Fic. 6.4. Space/time blocks produced for the quantity of interest equal to the average solution pver the whole domain Qy U Qe.
Fic. 6.4. Space/time blocks produced for the quantity of interest equal to the average solution pver the whole domain Qy U Qe.
Estimated total errors Eest and Eest computed for a quantity of interest equal to the average over the whole domain using the single-step (multiple refinements at once) and the iterative strategies (single refinement per iteration). DOF is the maximum number of degrees of freedom in a temporal block. ae es Estimated total errors Eest and Eecst computed for a quantity of interest equal to the averag over the whole domain on uniformly refined meshes. DOF is the total number of degrees of freedon in a mesh, Nz is the total number of time steps.
Estimated total errors Eest and Eest computed for a quantity of interest equal to the average over the whole domain using the single-step (multiple refinements at once) and the iterative strategies (single refinement per iteration). DOF is the maximum number of degrees of freedom in a temporal block. ae es Estimated total errors Eest and Eecst computed for a quantity of interest equal to the averag over the whole domain on uniformly refined meshes. DOF is the total number of degrees of freedon in a mesh, Nz is the total number of time steps.
The absolute error Es: only reaches the tolerance TOL = 0.005 on mesh 4, i.e., after three uniform refinements of the original mesh. The reason for the rapid decay of the actual error €¢s; with uniform refinement is the cancelation of errors due to the symmetry of the problem. This cannot be guaranteed, and will not necessarily occur in other problems.
The absolute error Es: only reaches the tolerance TOL = 0.005 on mesh 4, i.e., after three uniform refinements of the original mesh. The reason for the rapid decay of the actual error €¢s; with uniform refinement is the cancelation of errors due to the symmetry of the problem. This cannot be guaranteed, and will not necessarily occur in other problems.
Subsection 6.2.2. Estimated total errors Eest and Eecst for the quantity of interest defined on the interface, the temporal and spatial components of the error E; and Ex, and the corresponding L2-error er calculated on the original uniform mesh and on the mesh obtained after a space-time refinement with NMAX = 550. DOF is the maximum number of degrees of freedom in a temporal block.
Subsection 6.2.2. Estimated total errors Eest and Eecst for the quantity of interest defined on the interface, the temporal and spatial components of the error E; and Ex, and the corresponding L2-error er calculated on the original uniform mesh and on the mesh obtained after a space-time refinement with NMAX = 550. DOF is the maximum number of degrees of freedom in a temporal block.
FIG. 6.5. Subsection 6.2.2. Space/time blocks and temporal refinements obtained for a quantity of interest equal to the weighted average solution on the interface [ given NMAX = 550.
FIG. 6.5. Subsection 6.2.2. Space/time blocks and temporal refinements obtained for a quantity of interest equal to the weighted average solution on the interface [ given NMAX = 550.
FIG. 6.6. Subsection 6.3(a). Space/time blocks and temporal refinements produced for the quantity of interest equal to the average solution over the whole domain 2Q4UQ2 given NMAX = 380, TOL = 0.01.
FIG. 6.6. Subsection 6.3(a). Space/time blocks and temporal refinements produced for the quantity of interest equal to the average solution over the whole domain 2Q4UQ2 given NMAX = 380, TOL = 0.01.
Subsection 6.3(a). Estimated total errors Eest and Eest for a quantity of interest equal to the average error in the domain, their temporal and spatial components E, and Ex, and the corresponding L2-error eq calculated on the original uniform mesh and on the mesh obtained after space-time refinement with NUAX = 380, TOL = 0.01. For both quantities of interest we observe some common features of the refined mesh. Fewer temporal blocks are required to trace the trajectory in 0, where the solution is now compressed relative to the case when k, = kg = 1, and travels closer to the interface. The highly refined region is noticeably thinner in the y-direction in the second block as compared to the other blocks. The error is reduced similarly to the kj = kp = 1 case. Note that in the case of interface refinements there is now less cancelation of error due to lack of symmetry in the solution, and the error € is higher than before. As before, temporal refinements are concentrated around t = 0.5
Subsection 6.3(a). Estimated total errors Eest and Eest for a quantity of interest equal to the average error in the domain, their temporal and spatial components E, and Ex, and the corresponding L2-error eq calculated on the original uniform mesh and on the mesh obtained after space-time refinement with NUAX = 380, TOL = 0.01. For both quantities of interest we observe some common features of the refined mesh. Fewer temporal blocks are required to trace the trajectory in 0, where the solution is now compressed relative to the case when k, = kg = 1, and travels closer to the interface. The highly refined region is noticeably thinner in the y-direction in the second block as compared to the other blocks. The error is reduced similarly to the kj = kp = 1 case. Note that in the case of interface refinements there is now less cancelation of error due to lack of symmetry in the solution, and the error € is higher than before. As before, temporal refinements are concentrated around t = 0.5
FIG. 6.7. Subsection 6.3(b). Space/time blocks and temporal refinements produced for the quantity of interest equal to the weighted average solution on the interface [ given NMAX = 400, TOL = 0.05.
FIG. 6.7. Subsection 6.3(b). Space/time blocks and temporal refinements produced for the quantity of interest equal to the weighted average solution on the interface [ given NMAX = 400, TOL = 0.05.
Subsection 6.3(b). Estimated total errors Eest and Eest for a quantity of interest defined o: the interface, their temporal and spatial components E; and Ez, and the corresponding L2-error e) calculated on the original uniform mesh and on the mesh obtained after space-time refinement wit NMAX = 400, TOL = 0.05.
Subsection 6.3(b). Estimated total errors Eest and Eest for a quantity of interest defined o: the interface, their temporal and spatial components E; and Ez, and the corresponding L2-error e) calculated on the original uniform mesh and on the mesh obtained after space-time refinement wit NMAX = 400, TOL = 0.05.
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