Moving spaces

• We identify a pair \(v = (v_1, v_2)\) for a function \(v\) defined on \(\Omega\) by \(v_i = v|_{\Omega_i}\).

• We define the spaces: \[ \begin{aligned} H(t) & = L^2(\Omega_1(t)) \times L^2(\Omega_2(t)) \\ V(t) & = \{ v \in H^1(\Omega_1(t)) \times H^1(\Omega_2(t)) \,|\, u_1|_{\Gamma(t)} = u_2|_{\Gamma(t)} \mbox{ and } u_2|_{\partial \Omega} = 0 \}. \end{aligned} \]

• \(V(t) \subset H(t) \subset V^*(t)\) form an evolving Hilbert triple.

(Alphonse et al., 2015)

Moving spaces (ii)

• Each spaces forms a compatible pair with the push-forward (or pull-back) map given by \[ \begin{aligned} \phi_t v &= (v_1(\Phi_{-t}), v_2(\Phi_{-t})) && \mbox{ for } v \in H(0) \\ \phi_{-t} v &= (v_1(\Phi_{t}), v_2(\Phi_{t})) && \mbox{ for } v \in H(t). \end{aligned} \]

• This allows us to define moving Sobolev spaces (for \(X = V, H\)) \[ L^2_X := \{ v \colon [0,T] \to \bigcup_{t \in [0,T]} X(t) \times \{ t \} \,|\, t \mapsto (\hat{v}(t), t), \phi_{-t} \hat{v}(t) \in L^2(0, T; X_0) \} \]

• and a material derivative for smooth function \(v\) by \[ \md_t v := \phi_t \left( \dt \phi_{-t} v \right) = \partial_t v + \vec{w} \cdot \nabla v. \]

(Alphonse et al., 2015)

The variational form

Well posedness

Discretisation of the domain (i)

• We assume we are given an initial (nice) piecewise linear triangulation \(\tilde{\T}_h\) of \(\Omega\) which approximates the domains at time \(t = 0\).

\[ \Psi_h(x) = x + (\mu^*(x))^{\textcolor{green}{k+2}} (p(y(x), 0) - y(x)) \quad \mbox{ if } \mu^*(x) \neq 0 \]

Discretisation of the domain (ii)

The initial mesh \(F_{K}(\hat{x}, 0)\) is defined by the interpolation of \(\Psi_h\) of order \(k\)

Time dependent domain

Lifted element

• We construct a bijection between the computational domain and the exact domain. We call this the lift denoted by \(\Lambda_h \colon \Omega \to \Omega\).

• We can also lift functions defined on the triangulation: \(v_h \mapsto v_h^\ell\).

• The process also defines a lifted material derivative. (Dziuk & Elliott, 2012)

• The procedure is similar to construction of the initial domain.

• Important: The lift is the identity mapping for all elements away from the interface.

Finite element spaces

\[ \begin{aligned} S_h^i(t) := \Big\{ & \chi_h = (\chi_K)_{K \in \T_h^i(t)} \in \prod_{K \in \T_h^i(t)} \{ \hat\chi \circ F^{-1}_{K} : \hat\chi \in P_k(\hat{K}) \} : \\ & \chi_K(a(t)) = \chi_{K'}(a(t)) \mbox{ for all } K, K' \in \T^i(a(t)), \\ & \qquad \mbox{ for all } a(t) \in \N_h^i(t) \Big\} \end{aligned} \]

\[ \begin{aligned} S_h(t) := \big\{ & \chi_h = (\chi^1_h, \chi^2_h) \in S_h^1(t) \times S_h^2(t) : \\ & \chi_h^1(a(t)) = \chi_h^2(a(t)) \mbox{ for all } a(t) \in \N_h^\Gamma(t), \\ & \chi_h^2(a) = 0 \mbox{ for all } a \in \N_h^{\partial \Omega} \big\} \end{aligned} \]

Discrete problem

Ideas of proof

• We apply the ideas from (Elliott & TR, 2020).

• We use the Ritz projection \(\Pi_h \colon V(t) \to S^h(t)\) given by \[ a_h(t; \Pi_h z, v_h) = a(t; z, v_h^\ell) \qquad \mbox{ for all } v_h \in S^h(t). \]

• Split errors: \[ u - U_h^\ell = \rho + \theta = (u - (\Pi_h u)^\ell) + (\Phi_h u - U_h)^\ell. \]

• Estimates for \(\rho\) and \(\theta\) both require geometric estimates, interpolation estimates, duality arguments.

• The geometric errors compare a lifted problem (with lifted material derivative) to the continuous problem.

Geometric estimates (example) i

We want to show: \[ \tag{P1} \abs{m(t; \eta_h^\ell, v_h^\ell) - m_h(t; \eta_h, v_h)} \le c h^{k+1} \norm{\eta_h^\ell}_{V(t)} \norm{v_h^\ell}_{V(t)}. \]

where

\[ \begin{aligned} m(t; \eta, v) &= \sum_i \int_{\Omega^i} \eta^i v^i \dd x \\ m_h(t; \eta_h, v_h) &= \sum_i \int_{\Omega_h^i} \eta_h^i v_h^i \dd x. \end{aligned} \]

Geometric estimates (example) ii

\[ \tag{P1} \abs{m(t; \eta_h^\ell, v_h^\ell) - m_h(t; \eta_h, v_h)} \le c h^{k+1} \norm{\eta_h^\ell}_{V(t)} \norm{v_h^\ell}_{V(t)}. \]

• Let \(J_h\) be the Jacobian \(\sqrt{\det( \nabla \Lambda_h^T \nabla \Lambda h)}\) then \[ \sup_{t \in [0, T]} \norm{J_h - 1}_{L^\infty(\Omega_{h,i}(t))} \le c h^k. \]

• \[ \begin{aligned} & \abs{m(t; \eta_h^\ell, v_h^\ell) - m_h(t; \eta_h, v_h)} = \abs{\int_{\Omega} \eta_h^\ell v_h^\ell (J_h - 1)} \\ & \qquad \le c h^k \norm{ \eta_h }_{L^2(\Omega)} \norm{ v_h }_{L^2(\Omega)}. \end{aligned} \]

Geometric estimates (example) iii

• Denote by \([J_j \neq 1]\) be the elements where \(J_h \neq 1\) then \[ \begin{aligned} & \abs{m(t; \eta_h^\ell, v_h^\ell) - m_h(t; \eta_h, v_h)} = \abs{\int_{\Omega} \eta_h^\ell v_h^\ell (J_h - 1)} \\ & \le c h^k \left(\int_{[J_h \neq 1]} (\eta_h^\ell)^2 \right)^{1/2} \left(\int_{[J_h \neq 1]} (v_h^\ell)^2 \right)^{1/2}. \end{aligned} \]

• Applying the narrow band trace inequality(Elliott & TR, 2012) we get the result: For any \(\eta \in H^1(\Omega)\) \[ \norm{ \eta }_{L^2([J_h \neq 1])} \le c h^{1/2} \norm{ \eta }_{H^1(\Omega)}. \]

• \[ \abs{m(t; \eta_h^\ell, v_h^\ell) - m_h(t; \eta_h, v_h)} \le c h^{k+1} \norm{ \eta_h }_{V(t)} \norm{ v_h }_{V(t)}. \]

Geometric estimates - what about \(a\) form?

\[ \tag{P2} \abs{a(t; \eta_h^\ell, v_h^\ell) - a_h(t; \eta_h, v_h)} \le c h^{k+1} \norm{\eta_h^\ell}_{Z_0(t)} \norm{v_h^\ell}_{Z_0(t)}. \]

• We can apply the same ideas as for \(m\) bilinear forms to get order \(k+1\) estimate.

• But this needs careful application to ensure that we can apply this for smooth functions.

• We do this by using our Ritz projection so that (P2) is only applied to smooth functions.

Additional duality argument

• Define the bilinear form \(b \colon V(t) \times V(t) \to \R\) by \[ \dt a(t; \eta, v) = a(t; \md_t \eta, v) + a(t; \eta, \md_t v) + b(t; \eta, v). \]

• Let \(z \in Z_k(t)\) and \(\eta = z - (\Pi_h z)^\ell\). We need to show \[ \abs{b(t; \eta, \zeta)} \le c \left( \norm{\eta}_{H(t)} + h \norm{\eta}_{V(t)} + h^{k+1} \norm{z}_{Z_k(t)} \right) \norm{\zeta}_{Z_0(t)}. \tag{B3} \]

• To show this we use:

  • integration by parts;
  • a nice boundary duality argument (Douglas & Dupont, 1973).

Convergence result