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Fundamental Concepts of Finite Element Method (FEM)

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Abstract

In this chapter the basic principles of FEM are systematically developed and reviewed for prototypical advection-dispersion equations (ADE’s). The different spatial and temporal discretization techniques are addressed. The important approximate solutions for the divergence and convective forms of ADE are carefully developed. Emphasis is given on adaptive solution strategies. Implicit and explicit time integration methods are reviewed and compared. It clearly shows the superiority of implicit strategies for the present problem classes, in particular automatic error-controlled predictor-corrector schemes are favorited. Upwind methods are thoroughly discussed and examined in comparison to the standard Galerkin-based FEM (GFEM). Stability and error analyses for the favorite schemes are presented in some detail. Techniques for solving the resulting matrix equations are discussed. Of important interest is the solution of the nonlinear equations by using Picard and Newton iteration techniques, which are embedded in adaptive time stepping strategies for solving transient problems. A particular focus is given on derived quantities, i.e., the evaluation of fluxes and balance quantities. It is shown that the FEM is locally conservative.

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Notes

  1. 1.

    A continuous space-time approximation can be expressed in the form

    $$\displaystyle{\phi (\boldsymbol{x},t) \approx \hat{\phi } (\boldsymbol{x},t) =\sum _{j}N_{j}(\boldsymbol{x},t)\,\phi _{j}}$$

    where the basis functions \(\boldsymbol{x},t\) have to be prescribed both in space \(\boldsymbol{x}\) and time t. It requires a finite element in space-time and increases the computational dimension, e.g., a transient 3D problem needs a 4D trial space.

  2. 2.

    Let \(\mathcal{L}\) be a differential operator of a PDE defined in Ω and let ϕ and ψ be two functions in the field of definition of \(\mathcal{L}\). The operator \(\mathcal{L}\) is said to be self-adjoint if identical to its own adjoint operator \({\mathcal{L}}^{{\ast}}\), i.e., \(\mathcal{L} = {\mathcal{L}}^{{\ast}}\), which must result from the integral statement

    $$\displaystyle{\int _{\varOmega }\mathcal{L}(\phi )\psi d\varOmega =\int _{\varOmega }\phi {\mathcal{L}}^{{\ast}}(\psi )d\varOmega + \mbox{ boundary integral terms}}$$
  3. 3.

    For instance, the non-self-adjoint ADE in form of (8.5) can be transformed to a self-adjoint problem by introducing the new operator [129, 132, 218, 427]

    $$\displaystyle{\bar{\mathcal{L}} =\varphi \, \mathcal{L}}$$

    where the function \(\varphi =\varphi (\boldsymbol{x})\) is chosen by

    $$\displaystyle{\varphi =\exp (\beta ),\quad \beta = -\frac{\boldsymbol{q} \cdot \boldsymbol{ x}} {\Vert \boldsymbol{D}\Vert } }$$

    assuming a dispersion tensor \(\boldsymbol{D}\) with D ij = 0 for i ≠ j. It yields the following variational functional

    $$\displaystyle{\mathcal{I} =\int _{\varOmega }{\Bigl [\tfrac{1} {2}\nabla \phi \cdot (\boldsymbol{D} \cdot \nabla \phi ) +{\Bigl (\acute{ \mathcal{R}}\frac{\partial \phi } {\partial t} + \frac{\vartheta +Q} {2} \phi - H - Q_{\phi w}\Bigr )}\phi \Bigr ]}\,\exp (\beta )d\varOmega -\int _{\varGamma }(\boldsymbol{D} \cdot \nabla \phi ) \cdot \boldsymbol{ n}\phi \,\exp (\beta )d\varGamma }$$

    to be extremized. However, its application is clearly restricted because values of exp(β) can become very large (small) for advection-dominated processes and the variational functional terms overflow (underflow) in practical computations [129, 485].

  4. 4.

    We know from (2.26) when two vectors in space are at right angles, their dot product is zero and the vectors are orthogonal. While vectors have only a limited number of entries, any real-valued function \(f(\boldsymbol{x})\) is characterized by infinite number of points within its domain of definition Ω. It is obvious to consider two functions \(f(\boldsymbol{x})\) and \(g(\boldsymbol{x})\) to be orthogonal, if the product \(f(\boldsymbol{x})g(\boldsymbol{x})\) ‘summed’ over all \(\boldsymbol{x}\) within the domain Ω results zero. Since the amount of \(\boldsymbol{x}\) covers infinite real numbers, the product \(f(\boldsymbol{x})g(\boldsymbol{x})\) has to be integrated. Hence, the analogy for the dot product is the inner product given by

    $$\displaystyle{(f,g) =\int _{\varOmega }f(\boldsymbol{x})g(\boldsymbol{x})d\varOmega }$$

    Then, the two functions are orthogonal if (f, g) = 0. It is evident that functions if defined in the L 2(Ω) space, cf. (8.25), e.g.,

    $$\displaystyle{\Vert f\Vert ={\Bigl (\int _{\varOmega }f{(\boldsymbol{x})}^{2}{d\varOmega \Bigr )}}^{\tfrac{1} {2} } < \infty,\quad \Vert g\Vert ={\Bigl (\int _{\varOmega }g{(\boldsymbol{x})}^{2}{d\varOmega \Bigr )}}^{\tfrac{1} {2} } < \infty }$$

    can be treated if they were vectors, where the Schwarz’s inequality holds

    $$\displaystyle{(f,g) \leq \Vert f\Vert \Vert g\Vert }$$

    or

    $$\displaystyle\begin{array}{rcl} \vert {(f,g)\vert }^{2} \leq (f,f)(g,g)& & {}\\ \end{array}$$

    The L 2(Ω)−norm corresponds to a measure of the size of a function, which is in direct analogy with the vector norm (2.11). The Schwarz’s inequality ensures that the expression

    $$\displaystyle{\cos \theta = \frac{(f,g)} {\Vert f\Vert \Vert g\Vert } }$$

    yields well-defined angles θ in space similar to the scalar product of vectors (2.24).

  5. 5.

    The weak statements (8.94) and (8.95) imply that the weighting functions w i belong to the H 0 1 functional space (8.23). On the other hand, the basis functions N i belong to the H 1 functional space, i.e., they do not vanish on Dirichlet boundaries: \(N_{i}\neq 0\;\mbox{ on}\;\varGamma _{D}\). Nevertheless, we may use WS in form of (8.94) and (8.95) with \(w_{i} = N_{i} \in {H}^{1}(\varOmega ),\; 1 \leq i \leq N_{\mathrm{P}}\), where we enforce at first a zero flux \((\phi \boldsymbol{q} -\boldsymbol{ D} \cdot \nabla \phi ) \cdot \boldsymbol{ n} \approx 0\) or \(-(\boldsymbol{D} \cdot \nabla \phi ) \cdot \boldsymbol{ n} \approx 0\) on Γ D in the original weak statements (8.47) and (8.54), respectively, and incorporate the actual Dirichlet (essential) BC’s afterwards via a direct manipulation of the resulting discrete matrix system as further discussed in Sect. 8.16.

  6. 6.

    We can alternatively write the matrices and vectors in using directly the global shape function (8.85) with the global node numbers i, j:

    $$\displaystyle\begin{array}{rcl} \begin{array}{rcl} O_{\mathit{ij}} & = & \left \{\begin{array}{ll} \sum _{e}\int _{{\varOmega }^{e}}{\mathcal{R}}^{e}\,N_{ i}N_{j}{d\varOmega }^{e}&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;\,\;\,\mbox{ divergence form} \\ \sum _{e}\int _{{\varOmega }^{e}}\acute{{\mathcal{R}}}^{e}\,N_{ i}N_{j}{d\varOmega }^{e}&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;\,\;\,\mbox{ convective form} \end{array} \right. \\ A_{\mathit{ij}} & = & \left \{\begin{array}{ll} -\sum _{e}\int _{{\varOmega }^{e}}\boldsymbol{{q}}^{e} \cdot \nabla N_{ i}N_{j}{d\varOmega }^{e}&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\,\mbox{ divergence form} \\ \sum _{e}\int _{{\varOmega }^{e}}N_{i}\boldsymbol{{q}}^{e} \cdot \nabla N_{ j}{d\varOmega }^{e} &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\,\mbox{ convective form} \end{array} \right. \\ C_{\mathit{ij}} & = & \sum _{e}\int _{{\varOmega }^{e}}\nabla N_{i} \cdot (\boldsymbol{{D}}^{e} \cdot \nabla N_{ j}){d\varOmega }^{e} \\ R_{\mathit{ij}} & = & \left \{\begin{array}{ll} \sum _{e}\int _{{\varOmega }^{e}}{(\vartheta }^{e} + \tfrac{\partial {\mathcal{R}}^{e}} {\partial t} )N_{i}N_{j}{d\varOmega }^{e} &\qquad \qquad \qquad \quad \;\;\;\;\,\mbox{ divergence form} \\ \sum _{e}\int _{{\varOmega }^{e}}{(\vartheta }^{e} + {Q}^{e})N_{ i}N_{j}{d\varOmega }^{e} -\delta _{\mathit{ ij}}Q_{w}(t)\vert _{i}&\qquad \qquad \qquad \qquad \;\mbox{ convective form} \end{array} \right. \\ B_{\mathit{ij}} & = & \left \{\begin{array}{ll} \sum _{e}{\Bigl (\int {_{\varGamma _{C}^{e}}\varPhi }^{{\dag }^{e}}N_{ i}N_{j}{d\varGamma }^{e} +\int _{\varGamma _{ N_{O}}^{e}}N_{i}(\boldsymbol{q}N_{j} -\boldsymbol{ D} \cdot \nabla N_{j}) \cdot \boldsymbol{ n}{d\varGamma }^{e}\Bigr )}&\mbox{ divergence form} \\ \sum _{e}{\Bigl (\int {_{\varGamma _{C}^{e}}\varPhi }^{e}N_{ i}N_{j}{d\varGamma }^{e} -\int _{\varGamma _{ N_{O}}^{e}}N_{i}(\boldsymbol{D} \cdot \nabla N_{j}) \cdot \boldsymbol{ n}{d\varGamma }^{e}\Bigr )} &\mbox{ convective form} \end{array} \right. \\ H_{i}& = & \left \{\begin{array}{ll} \sum _{e}{\Bigl (\int _{\varGamma _{C}^{e}}N{_{i}\varPhi }^{{\dag }^{e}}\phi _{ C}^{e}{d\varGamma }^{e} -\int _{\varGamma _{ N}^{e}\setminus \varGamma _{N_{ O}}^{e}}N_{i}q_{N}^{{\dag }^{e}}{d\varGamma }^{e}\Bigr )}&\qquad \qquad \quad \;\;\;\,\;\,\mbox{ divergence form} \\ \sum _{e}{\Bigl (\int _{\varGamma _{C}^{e}}N{_{i}\varPhi }^{e}\phi _{ C}^{e}{d\varGamma }^{e} -\int _{\varGamma _{ N}^{e}\setminus \varGamma _{N_{ O}}^{e}}N_{i}q_{N}^{e}{d\varGamma }^{e}\Bigr )} &\qquad \qquad \quad \;\;\;\,\;\,\mbox{ convective form} \end{array} \right. \\ Q_{i}& = & \sum _{e}\int _{{\varOmega }^{e}}N_{i}{H}^{e}{d\varOmega }^{e} -\phi _{w}Q_{w}(t)\vert _{ i} \end{array} & & {}\\ \end{array}$$
  7. 7.

    Note that:

    $$\displaystyle\begin{array}{rcl} \begin{array}{rcl} \boldsymbol{{q}}^{e} \cdot \nabla N_{I}^{e}N_{J}^{e}& = & {\bigl (N_{I}^{e}\boldsymbol{{q}}^{e} \cdot \nabla N_{J}^{{e}\bigr )}}^{T} \\ \boldsymbol{{q}}^{e} \cdot \nabla N_{i}N_{j}& = & {\bigl (N_{i}\boldsymbol{{q}}^{e} \cdot \nabla N{_{j}\bigr )}}^{T} \end{array} & & {}\\ \end{array}$$
  8. 8.

    The local coordinates \(\boldsymbol{\eta }\) commonly range \(-1 \leq \boldsymbol{\eta }\leq +1\), except in triangular or tetrahedral (and partly pentahedral and pyramidal) geometries, where the lower limit is zero: \(0 \leq \boldsymbol{\eta }\leq +1\), see furthermore (8.128).

  9. 9.

    The 2nd-order derivatives are obtained by repeated application of 1st-order approximations:

    $$\displaystyle{\ddot{\boldsymbol{\phi }}_{n+1} = \frac{\dot{\boldsymbol{\phi }}_{n+1} -\dot{\boldsymbol{\phi }}_{n}} {\varDelta t_{n}} + \mathcal{O}(\varDelta t_{n}),\quad \ddot{\boldsymbol{\phi }}_{n} = \frac{\dot{\boldsymbol{\phi }}_{n+1} -\dot{\boldsymbol{\phi }}_{n}} {\varDelta t_{n}} -\mathcal{O}(\varDelta t_{n})}$$

    so that

    $$\displaystyle{\ddot{\boldsymbol{\phi }}_{n+1} =\ddot{\boldsymbol{\phi }} _{n} + \mathcal{O}(\varDelta t_{n})}$$
  10. 10.

    Truncated Taylor series expansions for \(\boldsymbol{\phi }_{n-1}\) and \(\boldsymbol{\phi }_{n+1}\) about t n give:

    $$\displaystyle{\boldsymbol{\phi }_{n-1} =\boldsymbol{\phi } _{n} -\varDelta t_{n-1}\dot{\boldsymbol{\phi }}_{n} + \frac{\varDelta t_{n-1}^{2}} {2} \ddot{\boldsymbol{\phi }}_{n},\quad \boldsymbol{\phi }_{n+1} =\boldsymbol{\phi } _{n} +\varDelta t_{n}\dot{\boldsymbol{\phi }}_{n} + \frac{\varDelta t_{n}^{2}} {2} \ddot{\boldsymbol{\phi }}_{n}}$$

    Using the first expression to write \(\dot{\boldsymbol{\phi }}_{n} = \frac{\boldsymbol{\phi }_{n}-\boldsymbol{\phi }_{n-1}} {\varDelta t_{n-1}} + \frac{\varDelta t_{n-1}} {2} \ddot{\boldsymbol{\phi }}_{n}\) and inserting into the second formula with \(\frac{\varDelta t_{n}} {2} \ddot{\boldsymbol{\phi }}_{n} = \frac{\boldsymbol{\phi }_{n+1}-\boldsymbol{\phi }_{n}} {\varDelta t_{n}} -\dot{\boldsymbol{\phi }}_{n}\), we obtain

    $$\displaystyle{\dot{\boldsymbol{\phi }}_{n} = \frac{\boldsymbol{\phi }_{n} -\boldsymbol{\phi }_{n-1}} {\varDelta t_{n-1}} + \frac{\varDelta t_{n-1}} {\varDelta t_{n}} {\biggl (\frac{\boldsymbol{\phi }_{n+1} -\boldsymbol{\phi }_{n}} {\varDelta t_{n}} -\dot{\boldsymbol{\phi }}_{n}\biggr )}}$$

    After some manipulations we finally find

    $$\displaystyle{\dot{\boldsymbol{\phi }}_{n} = \frac{\varDelta t_{n-1}} {\varDelta t_{n} +\varDelta t_{n-1}}{\biggl (\frac{\boldsymbol{\phi }_{n+1} -\boldsymbol{\phi }_{n}} {\varDelta t_{n}} \biggr )} + \frac{\varDelta t_{n}} {\varDelta t_{n} +\varDelta t_{n-1}}{\biggl (\frac{\boldsymbol{\phi }_{n} -\boldsymbol{\phi }_{n-1}} {\varDelta t_{n-1}} \biggr )}}$$
  11. 11.

    Since \(O_{\mathit{ii}}^{e} = \frac{\acute{\mathcal{R}}\varDelta x} {3} \bigl |_{\mathrm{CM}}\) for a consistent mass (CM) matrix, the critical time step becomes even smaller:

    $$\displaystyle{\varDelta t_{n} <\varDelta t_{n}^{\mathrm{crit}} \approx \frac{\acute{\mathcal{R}}\varDelta {x}^{2}} {3D} }$$
  12. 12.

    The analytical (exact) solution of a 1D ADE is [71], p. 388, [540], cf. also Sect. 12.5.1

    $$\displaystyle{\phi (x,t) =\phi _{0} + \tfrac{1} {2}(\phi _{D} -\phi _{0}){\biggl [\mathrm{erfc}{\biggl (\frac{x - {q}^{{\ast}}t} {2\sqrt{{D}^{{\ast} } t}}\biggr )} +\exp {\Bigl ( \frac{x{q}^{{\ast}}} {{D}^{{\ast}}}\Bigr )}\;\mathrm{erfc}{\biggl (\frac{x + {q}^{{\ast}}t} {2\sqrt{{D}^{{\ast} } t}}\biggr )}\biggr ]}}$$

    valid for the IC: ϕ(x, 0) =ϕ 0, and BC’s: ϕ(0, t) =ϕ D and \(\frac{\partial \phi } {\partial x}(\infty,t) = 0\), where

    $$\displaystyle{\mathrm{erfc}(a) = \tfrac{2} {\sqrt{\pi }}\int _{a}^{\infty }\exp ({-\xi }^{2})d\xi }$$

    is the complementary error function [71], ϕ 0 = 0 is the used initial value and ϕ D = 1 is the used Dirichlet-type BC at x = 0. Note that for evaluating the analytical exp(. )erfc(. ) expression the more suitable exf(. , . ) function is applied which will be further discussed in Sect. 12.5.1.

  13. 13.

    For the ADE convective form the continuous weighting functions (8.218) and the discontinuous weighting functions (8.220) lead to the same result. However, for the ADE divergence form only the continuous weighting functions (8.218) are applicable, where the element advection matrix \(\boldsymbol{{A}}^{e}\) (8.104) becomes

    $$\displaystyle\begin{array}{rcl} \boldsymbol{{A}}^{e} = \tfrac{{q}^{e}} {2} \left (\begin{array}{cc} 1+\alpha & \;\;1-\alpha \\ - 1-\alpha &\;\; - 1+\alpha \end{array} \right )& & {}\\ \end{array}$$
  14. 14.

    A similar expression can be obtained for the ADE divergence form.

  15. 15.

    The ADE divergence form (8.3) contains a divergence expression of the advection term in form of \(\nabla \cdot (\boldsymbol{q}\phi )\). For the split advective part (8.267) the advective operator \({\mathcal{L}}^{a}\) would be

    $$\displaystyle{{\mathcal{L}}^{a} =\boldsymbol{ q} \cdot \nabla + (\nabla \cdot \boldsymbol{ q})}$$

    Then, the LS weak statement of the advective part is

    $$\displaystyle{\mathrm{LSWS} =\int _{\varOmega }{\bigl [\acute{\mathcal{R}}N_{i} +\theta \varDelta t_{n}\nabla \cdot (\boldsymbol{q}N_{i})\bigr ]}{\bigl [\acute{\mathcal{R}}(\dot{\phi }{-\dot{\phi }}^{d}) + \nabla \cdot (\boldsymbol{q}\phi )\bigr ]}d\varOmega = 0}$$

    which leads to a matrix system equivalent to (8.284), but having different element matrices

    $$\displaystyle{\begin{array}{rcl} \boldsymbol{{A}}^{e}& = & \int _{{\varOmega }^{e}}N_{I}^{e}[(\boldsymbol{{q}}^{e} \cdot \nabla N_{ J}^{e}) + (\nabla \cdot \boldsymbol{ {q}}^{e})N_{ J}^{e}]{d\varOmega }^{e} \\ \boldsymbol{{V }}^{e}& = & \int _{{\varOmega }^{e}}[(\boldsymbol{{q}}^{e} \cdot \nabla N_{ I}^{e}) + (\nabla \cdot \boldsymbol{ {q}}^{e})N_{ I}^{e}]N_{ J}^{e}{d\varOmega }^{e} \\ \boldsymbol{{T}}^{e}& = & \int _{{\varOmega }^{e}}\varDelta t_{n} \tfrac{1} {\acute{\mathcal{R}}}[\nabla \cdot (\boldsymbol{{q}}^{e}N_{ I}^{e})][\nabla \cdot (\boldsymbol{{q}}^{e}N_{ J}^{e})]{d\varOmega }^{e} \end{array} }$$

    While the symmetry of the matrix system is still maintained since \(\boldsymbol{{A}}^{e} ={ \boldsymbol{{V }}^{e}}^{T}\), the divergence expressions \((\nabla \cdot \boldsymbol{ {q}}^{e})\) appearing in \(\boldsymbol{{A}}^{e}\), \(\boldsymbol{{V }}^{e}\) and \(\boldsymbol{{T}}^{e}\) can cause difficulties if the flow is not selenoidal (i.e., not divergence-free: \(\nabla \cdot \boldsymbol{ q}\neq 0\)) at the presence of storage and sources/sinks. This makes the LS technique rather inappropriate for the ADE divergence form.

  16. 16.

    If the Jacobian \(\boldsymbol{J}{(\boldsymbol{\phi }}^{\tau }) = \partial \boldsymbol{R}{(\boldsymbol{\phi }}^{\tau })/{\partial \boldsymbol{\phi }}^{\tau }\) is not analytically available or too difficult for an analytical evaluation, it can be constructed numerically via a secant approximation by using a possibly very small increment δ in a form such as

    $$\displaystyle{\boldsymbol{J}{(\boldsymbol{\phi }}^{\tau }) \approx \frac{\boldsymbol{R}{(\boldsymbol{\phi }}^{\tau }+\delta ) -\boldsymbol{ R}{(\boldsymbol{\phi }}^{\tau })} {\delta } }$$

    The increment δ should not be chosen too small to avoid roundoff errors. On the other hand, a too large δ leads to a poor approximation of the Jacobian. A reasonable choice is the square root of the unit roundoff being about \(\epsilon _{R} = 1{0}^{-12}\) in double precision arithmetic, accordingly \(\delta = \sqrt{\epsilon _{R}} = 1{0}^{-6}\). The extra effort of the numerical evaluation consists of additional N EQ evaluation of residual \(\boldsymbol{R}\).

  17. 17.

    Superconvergence of the derivatives can be shown for the Gauss points, at least for quadrilateral elements [590]. On the other hand, the location of the superconvergent points for triangular elements is not fully known. Zienkiewicz and Zhu [594] propose to use optimal points, for instance the central points for linear triangles.

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Diersch, HJ.G. (2014). Fundamental Concepts of Finite Element Method (FEM). In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_8

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