PAMM · Proc. Appl. Math. Mech. 14, 839 – 840 (2014) / DOI 10.1002/pamm.201410400 Space–time DG methods for the coupled electro–mechanical activation of the human heart Elias Karabelas 1, * and Olaf Steinbach 1 1 Institut für Numerische Mathematik, TU Graz, Steyrergasse 30, 8010 Graz, Austria We consider the coupled system of time–dependent nonlinear partial differential equations modeling the electromechanical response of human heart tissue. Instead of time–stepping schemes we use a discontinuous Galerkin finite element method in the space–time domain to be able to resolve the solution in space and time simultaneously. c  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Model problem We consider the system of partial differential equations modeling the electromechanical activation of human heart tissue where the coupling is done via the active stress [1–3], i.e., in the space–time domain Q T := Ω × (0,T ) we have to find the transmembrane potential V tm , the extracellular potential u e , and the displacement field U such that ∂ ∂t V tm + I ion (V tm , v, U ) - Div(M i Grad V tm ) - Div (M i Grad u e )= s i , (1) - Div(M i Grad V tm ) - Div((M i + M e ) Grad u e )=0, (2) ∂ ∂t v + H(V tm ,v)=0, ∂ ∂t S a + G(V tm ,S a )=0, (3) Div(F(U )(S pas (U )+ S act (S a , U ))) = 0, (4) and we impose initial conditions and homogeneous Neumann boundary. Recall that C = F > F is the right Cauchy–Green strain tensor, and F := I + Grad U . For a hyperelastic material we have S pas := 2 ∂Ψ(C) ∂C where the Helmholtz free energy Ψ is taken as in [4]. The active stress is given by S act := S a (Cf 0 , f 0 ) - 1 /2 (f 0 ⊗ f 0 ), where f 0 denotes the muscle fiber direction. M i and M e are anisotropic conductivity tensors, and I ion and H are given functions describing the ionic current exchange. 2 Space–time discontinuous Galerkin finite element discretization In contrast to other approaches we consider a decomposition of the space–time domain Q T into finite elements, i.e. pentatopes in the three-dimensional spatial case [5]. By using a discontinuous Galerkin finite element method to discretize the linearized coupled system (1)–(4) we arrive at b DG (δV tm ,φ)+ a DG i (δV tm ,φ)+ a DG i (δu e ,φ)+ d Iion,k Vtm (δV tm ,φ)+ d Iion,k v (δv, φ)+ d Iion,k U (δU ,φ)=r DG 1 (φ), (5) a DG i (δV tm ,ψ)+ a DG i+e (δu e ,ψ)=r DG 2 (ψ), (6) b DG (δv, ζ )+ d H,k Vtm (δV tm ,ζ )+ d H,k v (δv, ζ )=r DG 3 (ζ ), (7) b DG (δS a ,η)+ d G,k Vtm (δV tm ,η)+ d G,k Sa (δS a ,η)=r DG 4 (η), (8) a DG elast (δU , τ )+ d elast,k Sa (δS a , τ )=r DG 5 (τ ), (9) where the bilinear forms a DG i (δV tm ,φ),a DG i+e (δu e ,φ) result from a standard symmetric interior penalty discontinuous Galerkin discretization of a heterogeneous diffusion equation [6, Ch. 4.5]. For discretizing the time derivative we use an upwind scheme as known for convection problems, b DG (u, v) := - N X l=1 Z τ l u∂ t v dq + X Γ kl ∈IE Z Γ kl {u} up kl JvK kl,t ds q + Z Σ T uv ds q , * Corresponding author, e-mail elias.karabelas@tugraz.at c  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim