\[ \newcommand{\abs}[1]{|#1|} \newcommand{\norm}[1]{\|#1\|} \newcommand{\A}{\mathscr{A}} \newcommand{\B}{\mathscr{B}} \newcommand{\C}{\mathscr{C}} \newcommand{\dt}{\frac{d}{dt}} \newcommand{\id}{\text{id}} \newcommand{\md}{\partial^\bullet} \DeclareMathOperator{\meas}{\mathrm{meas}} \newcommand{\Nbb}{\mathbb{N}} \newcommand{\P}{P} \newcommand{\R}{\mathbb{R}} \newcommand{\S}{\mathcal{S}} \newcommand{\T}{\mathcal{T}} \newcommand{\X}{X} \]

A unified theory for continuous in time evolving finite element space approximations to partial differential equations in evolving domains

Thomas Ranner

Preprint available arXiv:1703.04679. Slides available at tomranner.org/enumath2019

Motivation

Our aims

  • Provide basic theory of how to construct and analyse finite element methods for parabolic problems in evolving domains
  • This includes problems in evolving bulk domains, on evolving surfaces and coupled bulk-surface problems
  • How to define and improve quality of moving meshes
  • How to generalise (piecewise linear/isoparametric) surface finite elements to higher order elements

Model problem

Given “smoothly evolving smooth surface” \(\Gamma(t)\) find a scalar field \(u\) such that \[ \begin{align*} \partial^\bullet u + u \nabla_\Gamma \cdot w - \Delta_\Gamma u & = 0 \\ u(\cdot, 0) & = u_0. \end{align*} \]

(or general parabolic form on evolving surface, evolving domain, arbitrary parameterisations, …)

Recap on abstract theory of (Alphonse, Elliott, and Stinner 2015)

(see also (Vierling 2014))

Compatibility

Let \(X(t)\) be a family of separable Hilbert spaces, \(\phi_t \colon X(0) \to X(t)\) homeomorphisms, the pair \((X(t), \phi_t)_{t \in [0,T]}\) are called compatible if \[ C_X^{-1} \| \phi_t \eta \|_{X(t)} \le \| \eta \|_{X(0)} \le C_X \| \phi_t \eta \|_{X(t)} \qquad \mbox{ for all } \eta \in X(0). \]

Consequences of compatibility

For a compatible pair \(( X(t), \phi_t )_{t \in [0,T]}\), we can define the Hilbert space \(L^2_X\): \[ \begin{multline*} L^2_\X := \Big\{ \eta \colon [0,T] \to \bigcup_{t \in [0,T]} \X(t) \times \{ t \}, t \mapsto ( \bar\eta(t), t ) : \\ \phi_{-\cdot} \bar\eta(\cdot) \in L^2( 0, T; \X_0 ) \Big\} \end{multline*} \] and strong material derivative \[ \md \eta := \phi_t \left( \dt (\phi_{-t} \eta) \right) \mbox{ for } \eta \in C^1_X. \]

Abstract formulation of the pde

Given an evolving Hilbert triple of compatible spaces \((V^*(t), H(t), V(t))\), find \(u \in L^2_V\) such that \[ m( t; \partial^\bullet u, \varphi ) + g( t; u, \varphi ) + a( t; u, \varphi ) = 0 \qquad \mbox{ for all } \varphi \in V(t). \]

Well posedness

Theorem. Under appropriate assumptions on the spaces, push-forward maps and bilinear forms, the continuous problem has a unique solution \(u\in L^2_V\) with \(\partial^\bullet u \in L^2_H\) which satisfies the stability bounds \[ \begin{align*} \sup_{t\in [0,T]} \| u \|_{H(t)}^2 + \int_0^T \| u \|_{V(t)}^2 \, d t & \le c(T) \| u_0 \|_{H_0}^2, \\ \sup_{t\in [0,T]} \| u \|_{V(t)}^2 + \int_0^T \| \partial^\bullet u \|_{H(t)}^2 \, d t & \le c(T) \| u_0 \|_{V_0}^2. \end{align*} \]

Evolving surface finite element spaces

Surface finite element reference map

Let \(( \hat{K}, \hat{P}, \hat{\Sigma} )\) be a reference finite element (i.e. standard) with \(\hat{K} \subset \R^n\). Let \(F_K\) satisfy:

\(F_K \in C^1( \hat{K}, \R^{n+1} )\)    • \(\text{rank} \nabla F_K = n\)    • \(F_K\) is injective,

and that \(F_K\) can be decomposed into an affine and smooth part: \[ F_K( \hat{x} ) = A_K \hat{x} + b_k + D_K( \hat{x} ) \] such that \(A_K\) has full column rank, \(D_K \in C^1( \hat{K} )\) and \[ C_K := \sup_{\hat{x} \in \hat{K}} \norm{ \nabla D_K( \hat{x} ) A_K^\dagger } < 1. \] In this case, we call \(F_K\) a surface finite element reference map.

Surface finite element

Let \(( \hat{K}, \hat{P}, \hat{\Sigma} )\) be a reference finite element (i.e. standard) with \(\hat{K} \subset \R^n\), \(F_K\) a surface finite element reference map, and the triple \((K, P, \Sigma)\) given by

  • \(K := F_K( \hat{K} )\) (the element domain)
  • \(P := \{ \hat\chi \circ F_K^{-1} : \hat{\chi} \in \hat{P} \}\) (the shape functions)
  • \(\Sigma := \{ \chi \mapsto \hat{\sigma}( \chi \circ F_K ) : \hat{\sigma} \in \hat\Sigma \}\) ( the nodal variables).

Then we call \((K, P, \Sigma)\) a surface finite element.

\(\Theta\)-surface finite element

If in addition

  • \(F_K \in C^{\Theta+1}( \hat{K}; \R^{n+1} )\)
  • for \(1 \le m \le \Theta+1\) \[ \sup_{\hat{x} \in \hat{K}} \abs{ \nabla^m F_K( \hat{x} ) } \norm{ A_K }^{-m} \le C_M( K ) \]
  • \(P\) contains \(\hat{\chi} \circ F_K^{-1}\) for all \(\hat{\chi} \in P_\Theta( \hat{K} )\)
  • \(P \subset C^{\Theta+1}( K )\)

then we call \((K, P, \Sigma)\) a \(\Theta\)-surface finite element.

see also (Bernardi 1989)

Examples

Element examples
  • reference element in \(\R^2\) (left)
  • affine surface finite elements (center) (Dziuk 1988)
  • isoparametric surface finite elements (right) (Heine 2005)

Norm scaling properties

If \(\chi \in W^{m,p}( K )\) then \(\hat\chi = \chi \circ F_K \in W^{m,p}(\hat{K})\) and \[ \abs{ \hat{\chi} }_{W^{m,p}( \hat{K} ) } \le c \meas(K)^{-1/p} \norm{ A_K }^m \sum_{r=1}^m \abs{ \chi }_{W^{r,p}(K)}. \] If \(\hat{\chi} \in W^{m,p}( \hat{K} )\) then \(\chi = \hat\chi \circ F_K^{-1} \in W^{m,p}( \hat{K} )\) and \[ \abs{ \chi }_{W^{m,p}( K )} \le c \meas(K)^{1/p} \sum_{r=1}^m \norm{ A_K^\dagger }^r \norm{ \hat\chi }_{W^{r,p}( \hat{K} ) }. \]

Interpolation property

If the reference element satisfies a Bramble-Hilbert Lemma then for all \(\chi \in W^{k+1,p}( K )\) \[ \abs{ \chi - I_K \chi }_{W^{m,p}(K)} \le c\, \text{meas}(K)^{1/q - 1/p} \frac{h_K^{k+1}}{\rho_K^m} \abs{ \chi }_{W^{k+1,p}(K)}. \]

How to bring several elements together?

We restrict to Lagrange finite elements: \[ \Sigma := \{ \chi \mapsto \chi( a ) : a \in N^K \}. \]

Let \(\Gamma_h\) be discrete surface with conforming subdivision \(\T_h\). We assume that for any two adjacent (i.e. which share a common facet) surface finite elements \((K, P, \Sigma)\) and \((K', P', \Sigma' )\) that \[ \left( \bigcup_{a \in N^K} a \right) \cap K' = \left( \bigcup_{a' \in N^{K'}} a' \right) \cap K. \]

Surface finite element space

Let each \(K \in \T_h\) have an associated surface finite element \((K, P^K, \Sigma^K)\). A surface finite element space is a (proper) subset of the product space \(\prod P^K\) given by \[ \begin{multline*} \S_h := \Big\{ \chi_h = ( \chi_K )_{K \in \T_h} \in \prod_{K \in \T_h} P^K : \\ \chi_K( a ) = \chi_{K'}( a ) \text{ for all } K, K' \in \T( a ), \text{ and all } a \in N_h \Big\}. \end{multline*} \]

Lemma. We can identify each \(\chi_h \in \S_h\) with a continuous function.

Examples

Piecewise linear function

Piecewise quadratic function

Evolving surface finite element

We say the family \((K(t), P(t), \Sigma(t))_{t \in [0,T]}\) is a evolving surface finite element if

  • each share a common reference element
  • \[ C_K := \sup_{t \in [0,T]} \sup_{\hat{x} \in \hat{K}} \norm{ D \Phi( \hat{x}, t ) A_K^\dagger(t) } < c < 1. \]

We say the family \((K(t), P(t), \Sigma(t))_{t \in [0,T]}\) is a \(\Theta\)-evolving surface finite element if

  • each \(( K(t), P(t), \Sigma(t) )\) is a \(\Theta\)-surface finite element
  • \[ \sup_{t \in [0,T]} C_m( K(t) ) \le c \le \infty \qquad \text{ for } 1 \le m \le \Theta+1. \]

Element flow map

There exists a family of maps \(\Phi^K_t \colon K_0 \to K(t)\) called the flow map given by \[ F_{K(t)} ( \hat{x} ) = \Phi^K_t( F_{K_0}( \hat{x} ) ). \]

The flow map defines the element velocity field by \[ W_K( \Phi^K_t(x), t ) = \dt \Phi^K_t( x ). \]

Element push forward map

The family of element push forward maps \(\phi^K_t\) is defined for \(\chi \colon K_0 \to \R\) by \(\phi^K_t \chi \colon K(t) \to \R\) where \[ \phi^K_t( \chi )( x ) = \chi( \Phi^K_{-t}( x ) ). \]

Compatibility

We say that \((K(t), P(t), \Sigma(t))_{t \in [0,T]}\) is temporally quasi-uniform if there exists \(\rho_K > 0\) such that \[ \inf \{ \rho_{K(t)} : t \in [0,T] \} \ge \rho_K h. \]

Lemma. If an \(\Theta\)-evolving surface finite element \((K(t), P(t), \Sigma(t))_{t \in [0,T]}\) is temporally quasi-uniform then \(( W^{m,p}(K(t)), \phi^K_t)_{t \in [0,T]}\) (for \(0 \le m \le \Theta\), \(p \in [1,\infty]\)) and \(( P(t), \phi^K_t )_{t \in [0,T]}\) form compatible pairs.

Evolving surface finite element space

We restrict that element flow maps coincide at Lagrange points: for all \(a_0 \in N_h(0)\) we have \[ \Phi^K_t( a_0 ) = \Phi^{K'}_t( a_0 ) \text{ for adjacent } K(t), K'(t). \]

An evolving surface finite element space \(\S_h(t)\) is a time-dependent family of surface finite element spaces consisting of evolving surface finite elements.

Compatibility

We define a global push forward map for \(\eta_h \colon \Gamma_{h,0} \to \R\) by \(\phi^h_t \eta_h \colon \Gamma_h(t) \to \R\) by \[ ( \phi^h_t \eta_h )|_{K(t)} = \eta_h \circ \Phi^K_{-t}. \]

Lemma. If \(\T_h(t)\) is a uniformly quasi-uniform subdivision then each element is temporally quasi-uniform. In particular \(( W^{m,p}( \T_h(t), \phi^h_t ) )_{t \in [0,T]}\) (for \(0 \le m \le \Theta+1\), \(p \in [1,\infty]\)) and \(( \S_h(t), \phi^h_t )_{t \in [0,T]}\) are compatible pairs.

Material derivatives

Let \(H_h( t ) := L^2( \T_h(t) )\). We have that \((L^2( \T_h(t)), \phi^K_t)_{t \in [0,T]}\) form a compatible pair

  • we can define the spaces \(L^2_{H_h}\) and \(C^1_{H_h}\)
  • we can define a discrete material derivative \[ \md_h \eta_h := \phi_t^h \left( \dt \phi^h_{-t} \eta_h \right). \]

Lemma. Denote by \(\{ \chi_j( \cdot, t )\}_{j=1}^N\) the set of global basis functions. Then \(\md_h \chi_j = 0\).

Relation to continuous problem

We don’t have time in this talk to go into details….

Relating these definitions to their continuous counterparts requires lifting operators:

  • \(\Lambda_K \colon K(t) \to \Gamma\)
  • \(\lambda_h \colon H_h(t) \to H(t)\).

This also provides an interpolation operator \(I_h \colon C( \Gamma ) \to \S_h^\ell\).

Application

Given smoothly evolving surface \(\Gamma(t)\) find a scalar field \(u\) such that \[ \partial^\bullet u + u \nabla_\Gamma \cdot w - \Delta_\Gamma u = 0 \\ u(\cdot, 0) = u_0. \]

See also (Kovács 2018).

Construction of initial discrete surface

At initial time use interpolation of normal projection operator to define initial surface finite element reference map. Examples shown for isoparametric elements \(k=1,2,3\).

Construction of evolving discrete surface

Construct discrete flow map as interpolation of smooth flow map: Lagrange points move with velocity \(w\).

Evolving surface finite element space

Let \(\S_h(t)\) be an isoparametric finite element space over \(\T_h(t)\) (Demlow 2009; Kovács 2018). Assume that the evolving subdivision \(\{ \T_h(t) \}_{t \in [0,T]}\) is uniformly quasi-uniform.

Proposition. The above construction defines \(\S_h(t)\) to be an evolving surface finite element space consisting of \(k\)-evolving surface finite elements and the corresponding pair \(( \S_h(t), \phi^h_t )_{t \in [0,T]}\) form a compatible pair.

Further analysis of finite element scheme

  • We also give details of stability and error analyses based on an abstract formulation.
  • The results give show quasi-optimal error bounds for a general parabolic problem on an evolving open domain, evolving surface and a coupled bulk surface problem.
  • I do not have time to give any further details…

Numerical results

Consider \(\Gamma_0 = S^2 \subset \R^3\) the unit sphere and define \(\Gamma(t)\) by the velocity: \[ w( x, t ) = \left( \frac{\cos(t) x_1}{8(1+1/4 \sin(t))}, 0, 0 \right). \] For solution we take \(u(x,t) = \sin(t) x_2 x_3\) and consider a general parabolic operator with \(\A = ( 1 + x_1^2 ) \id\), \(\B = ( 1,2,0) - ( 1,2,0) \cdot \nu \nu\), \(\C = \sin(x_1 x_2)\)

Choose \(k=3\), time step to recover optimal scaling.

Numerical results

\(h\) \(\tau\) \(L^2(\Gamma(T))\) error (eoc)
\(8.31246\cdot10^{-1}\) \(1.00000\) \(9.88086\cdot10^{-2}\)
\(4.40053\cdot10^{-1}\) \(6.25000\cdot10^{-2}\) \(7.60635\cdot10^{-3}\) \(4.03157\)
\(2.22895\cdot10^{-1}\) \(3.90625\cdot10^{-3}\) \(4.92316\cdot10^{-4}\) \(4.02476\)
\(1.11969\cdot10^{-1}\) \(2.44141\cdot10^{-4}\) \(3.08257\cdot10^{-5}\) \(4.02448\)
\(5.60891\cdot10^{-2}\) \(1.52588\cdot10^{-5}\) \(1.89574\cdot10^{-6}\) \(4.03416\)

Summary

  • We have shown fundamental definitions of an evolving surface finite element space
  • We have (briefly) considered one particular realisation using isoparametric surface finite elements
  • We have given ideas of how one might generalise this theory
  • We have not shown how this relates to an abstract finite element analysis to show stability and error bounds
  • We have not shown similar constructions for problems in evolving bulk domains

Thank you for your attention! Preprint available arXiv:1703.04679. Slides available at tomranner.org/enumath2019

References

Alphonse, A., C. M. Elliott, and B. Stinner. 2015. “An Abstract Framework for Parabolic PDEs on Evolving Spaces.” Port. Math. 72 (1): 1–46. https://doi.org/10.4171/PM/1955.

Bernardi, C. 1989. “Optimal Finite-Element Interpolation on Curved Domains.” SIAM Journal on Numerical Analysis 26 (5): 1212–40. https://doi.org/10.1137/0726068.

Demlow, A. 2009. “Higher-order finite element methods and pointwise error estimates for elliptic problems on surface.” SIAM Journal on Numerical Analysis 47 (2): 805–27. https://doi.org/10.1137/070708135.

Dziuk, G. 1988. “Finite Elements for the Beltrami operator on arbitrary surfaces.” In Partial Differential Equations and Calculus of Variations, edited by Stefan Hildebrandt and Rolf Leis, 1357:142–55. Lecture Notes in Mathematics. Berlin: Springer-Verlag. https://doi.org/10.1007/BFb0082865.

Heine, C. J. 2005. “Isoparametric finite element approximation of curvature on hypersurfaces.” Freiburg. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.574.7294&rep=rep1&type=pdf.

Kovács, B. 2018. “High-Order Evolving Surface Finite Element Method for Parabolic Problems on Evolving Surfaces.” IMA Journal of Numerical Analysis 38: 430–59. https://doi.org/10.1093/imanum/drx013.

Vierling, M. 2014. “Parabolic optimal control problems on evolving surfaces subject to point-wise box constraints on the control – theory and numerical realization.” Interfaces and Free Boundaries 16 (2): 137–73. https://doi.org/10.4171/IFB/316.