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1.1	Complete the heat conduction example ODE solutions for given BCs, for each of the given heat source forms, IM.1.
.2	(Optional, brain-teaser): Resolve the lecture 1D unsteady heat conduction PDE, (IM.7) to determine the temperature distribution in a rod initially at uniform temperature T1, with the "right" end increased to T2 at time t0. (Hint: cast the solution process on the difference between the steady state temperature (linear in x) and the time-evolution of temperature.)
.3	(Optional, straight forward): Solve the 2D laplacian for on the rectangular domain 0<x<a, 0<y<b for the BCs (0,y)=0, grad(x,0).n = 0 = grad(x,b).n and (a,y) = cos(3y/b)	Due Class 5
2.1	Confirm the truncation error order for the second order accurate FD approximation for a first derivative, in one-dimension,(IM.26).
.2	Repeat the FD process  of 2.1 for a non-uniform mesh, in one dimension, hence determine the specific form of the truncation error.
.3	Derive the uniform mesh FD stencil for the Laplacian in 2D and 3D, c.f. (IM.30).
.4	Work through, hence verify the form (IM.41) for a Lagrange interpolation polynomial of degree K. Comment on potential non-uniqueness for K>1.	Due Class 5
3.1	Verify the analytical solution to the model problem on HC.2.
.2	Verify that the and Q cited on HC.2 for TN can agree with T(x).
.3	Verify the integrated-by-parts WS form, HC.4.
.4	For the linear basis, verify the matrix terms for GWSh, HC.7.
.5	Verify the assembly of components on HC.8 to form [Matrix] and {b}, HC.9.
.6	Verify the approximate solution {Q} on HC.10.
.7	Confirm the boundary flux solutions for F3, HC.12.
.8	Compute the FE GWSh solution for a 3-element non-uniform discretization of the span L of the problem statement. Observe, hence comment on nodal solution and flux accuracies.	Due Class 6
4.1	For the heat transfer problem statement on HT1.3, proceed through the WS steps leading to GWSh, (HT1.7).
.2	Verify the FE linear basis master matrix set on HT1.5.
.3	Verify the linear basis, 2-element uniform mesh solution, HT1.6.
.4	Verify the FE basis entries for {Nk} for k = 2, (HT1.21). Confirm that the entries exhibit the nodal (0,1) character for consistency.
.5	Verify that the FE basis {Nk} for k = 3, (HT1.22), exhibits the nodal (0,1) consistency check.
.6	Verify three of the matrix terms in [DIFF]e formed using {N2}, (HT1.30).
.7	Generate one matrix term in [DIFF]e formed using {N3}.
.8	Derive the truncation error estimate eh/2, (HT1.37).
.9	Derive the convergence slope equation, (HT1.38).
.10	For the non-linear heat transfer problem, HT1.23, proceed through the GWSh steps, hence verify the ingredients of the Newton algorithm, HT1.23.	Due Class 10
5.1	For the Euler-Bernoulli biharmonic beam equation, (CM1.1), proceed through the GWSN process leading to the template form (CM1.6).
.2	Verify that the entries in the cubic Hermite basis, (CM1.8), satisfy the nodal (0,1) consistency check.
.3	Verify the GWSh pair, (CM1.16) - (CM1.17), including BCs, for the Euler-Bernoulli biharmonic beam equation resolved into 2 PDEs, (CM1.15), hence the template on CM1.9.
.4	Verify the Euler-Bernoulli GWSh for non-uniform cross-section, (CM1.23), and the associated template (CM1.27) - (CM1.28).
.5	For the Timeshenko beam formulation, CM1.11, proceed through the GWSh process to confirm the solution template (CM1.33). Confirm the artificial diffusion term (CM1.32) resulting from under-integration.
.6	Verify the E-B beam harmonic vibration development (CM1.33) - (CM1.36), hence the template (CM1.38).
.7	Using the analytical integration formula (HT1.28), confirm a few matrix elements in [A200H]e, (CM1.39).
.8	Construct a couple terms in the element matrix [A3000LH]e for I(x) interpolation via {N1(z)}.
.9	For quasi-one dimensional potential flow, confirm the GWSh template (CM1.52) including the Neumann BC (CM1.51).
.10	Verify the template expressions for pressure and velocity post-processing, (CM1.55) and (CM1.57).
.11	For heat transfer in a finned cylinder, proceed through the GWSh process for (CM1.63) - (CM1.64), hence verify the template (CM1.68).
.12	For addition of the radiation heat transfer BC (CM1.70), verify the Newton template (CM1.71) - (CM1.72).	Due Class 13
6.0	Note: the use of symbolic software (e.g., Maple) is suggested for completion of some of the problems in this section.
.1	For triangles, verify the solution for a1, a2, a3 , (HTn.20).
.2	For tetrahedra, highlight the solution process for the ai - di, leading to (HTn.26), then verify the (0,1) nodal character of the FE linear basis {N1(i)}, (HTn.25).
.3	Verify the natural coordinate transformation matrix [i/xj], (HTn.31).
.4	Detail the construction of the n=2 diffusion matrix [DIFF]e, (HTn.37).
.5	Verify {b}e for a distributed source se, Table HTn.1.
.6	For quadrilaterals and hexahedra, verify that the {N1+}, (HTn.12), satisfy the (0,1) nodal requirements for an FE basis.
.7	Verify that any two members of {N2+}, (HTn.13), satisfy the nodal (0,1) conditions required of an FE basis.
.8	For quadrilaterals, verify the coordinate transformation (nj/xi), (HTn.47).
.9	Confirm the forms for the n=3,2 BC matrices, HTn.27, see also Table HTn.3.	Due Class 17
7.1	Verify that assembly Se of the FE linear natural coordinate approximation to the laplacian, for any cartesian triangle (tetrahedron) arrangement, reproduces the second order accurate FD form, FDV.3.
.2	Confirm the FV laplacian stencil (FDV.9) on the union of cartesian quadrilaterals.	Due Class 20
8.1	Using FDs, verify the truncation error terms for the  one-step time integration family, (CD1.17), for  = 0.0, 0.5 and 1.0.
.2	Verify the k=1 basis FE GWSh algorithm templates for n=1 linear, steady, and non-linar unsteady convection/diffusion, CD1.6 - 1.8.
.3	Verify the non-dimensionalization of (CD1.1) yielding (CD1.16), hence (CD1.35). Verify the analytical solution (CD1.36) for given Dirichlet (0,1) BCs.
.4	Determine the "Peclet problem" analytical solution energy semi-norm, i.e. the integral of 1/(2Pe) times (dq/dx)2, and determine its limit as Pe becomes very large.	Due Class 22
9.1	Confirm the n-D GWSh+qTS FE implementation template essence (CDn.15) - (CDn.17).
.2	For the n-D convection contribution (CDn.18), confirm the jacobian addition template (CDn.19).
.3	Re-express the GWSh+qDt template given in (CDn.22) for qDt and (1-q)Dt data definitions, giving specific attention to the source term.
.4	Develop the template replacement for {FQ}e, (CDn.22), for natural coordinate basis implementation.
.5	Verify the TS expansion manipulations given in (CDn.28), hence confirm Lm(q) as given in (CDn.31).
.6	Confirm the template for the Newton and AF TP quasi-Newton jacobians, (CDn.53) and (CDn.54).	Due Class 24
10.1	From first principles, develop one of the linear elasticity plane stress-plane strain PDE systems, (CMn.10) - (CMn.11), hence verify the vector form (CMn.12).
.2	Detail construction of the matrix PDE GWSN for plane stress/strain, (CMn.17).
.3	Verify the discrete extremization of the FE implementation of Pe (CMn.23). Then compare term by term with the matrix GWSN, (CMn.15), after FE implementation.
.4	Verify the construction of the [B] matrix, (CMn.29), for basis {N1(z)}.
.5	Compute several terms in the matrix product (CMn.31).
.6	Compare select matrix elements of [STIFF]e, (CMn.36), to [DIFF]e, (HTn.35).
.7	Expanding the curl in (CMn.43) via its determinant definition, confirm that (CMn.41) is identically satisfied.
.8	Substitute (CMn.43) into (CMn.44), hence confirm (CMn.46).
.9	Generate (CMn.49) from the TS on stream-function using (CMn.48).
.10	Confirm the forms for the lower order TS expressions in (CMn.51).
.11	Write the template statements for the Newton jacobian (CMn.56), using (CMn.54) and (CMn.57).
.12	Confirm (CMn.70) via the Euler-Lagrange equation definition.	Due Class 28
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