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

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)

## 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

• $F_K \in C^{\Theta+1}( \hat{K}; \R^{n+1} )$
• 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.

## 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

## 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

## 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). SIAM: 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.