In this work we use algebraic dual spaces with a domain decomposition method to solve the Darcy equations. We define the broken Sobolev spaces and their finite dimensional counterparts. A global trace space is defined that connects the solution between the broken spaces. Use of algebraic dual spaces results in a sparse, metric-free representation of the incompressibility constraint, the pressure gradient term, and on the continuity constraint between the sub domains. To demonstrate this, we solve two test cases: (i) a manufactured solution case, and (ii) an industrial benchmark reservoir modelling problem SPE10. The results demonstrate that the dual spaces can be used for domain decomposition formulation, and despite having more unknowns, requires less simulation time compared to the continuous Galerkin formulation, without compromising on the accuracy of the solution.

We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the sense that the finite dimensional function spaces form a discrete de Rham complex. Conservation of mass, kinetic energy and helicity in the absence of dissipative terms is proven at the discrete level. Proper dissipation rates of kinetic energy and helicity in the viscous case are also proven. Numerical tests supporting the method are provided.

A new horizontally explicit/vertically implicit (HEVI) time splitting scheme for atmospheric modelling is introduced, for which the horizontal divergence terms are applied within the implicit vertical substep. The new HEVI scheme is implemented in conjunction with a mixed mimetic spectral element spatial discretisation and semi-implicit vertical time stepping scheme that both preserve the skew-symmetric structure of the non-canonical Hamiltonian form of the equations of motion. Within this context the new HEVI scheme allows for the exact balance of all energetic exchanges in space and time. However since the choice of horizontal fluxes for which this balance is satisfied is not consistent with the horizontal velocity at the end of the time level the scheme still admits a temporal energy conservation error. Linearised eigenvalue analysis shows that similar to a fully implicit method, the new HEVI scheme is neutrally stable for all buoyancy modes, and unlike a second order trapezoidal HEVI scheme is stable for all acoustic modes below a certain horizontal CFL number. The scheme is validated against standard test cases for both planetary and non-hydrostatic regimes. For the planetary scale baroclinic instability test case, the new formulation exhibits a secondary oscillation in the potential to kinetic energy power exchanges, with a temporal frequency approximately four times that exhibited by a horizontally third order, vertically second order trapezoidal scheme. For the non-hydrostatic test case, the vertical upwinding of the potential temperature diagnostic equation is shown to reduce spurious oscillations without altering the energetics of the solution, since this upwinding is performed in an energetically consistent manner. For this test case, which is configured on an affine geometry, the exact balance of energy exchanges allows the model to run stably without any form of dissipation.

In ℝⁿ, let Λ^k(Ω) represent the space of smooth differential k-forms in Ω. The de Rham complex consists of a sequence of spaces, Λ^k(Ω), k = 0, 1…, n, connected by the exterior derivative, d: Λ^k(Ω) → Λ^k+1(Ω). Appropriately chosen B-spline spaces together with their associated dual B-spline spaces form a discrete double de Rham complex. In practical applications, this discrete double de Rham complex leads to very sparse systems. In this paper, this construction will be explained and illustrated by means of a non-trivial three-dimensional example.

In this paper, we present a hybrid mimetic method which solves the mixed formulation of the Poisson problem on curvilinear quadrilateral meshes. The method is hybrid in the sense that the domain is decomposed into multiple disjoint elements and the interelement continuity is enforced using a Lagrange multiplier. The method is mimetic in the sense that the discrete divergence operator is exact. By using the mimetic basis functions and their dual representations, various metric-free discrete terms are obtained. The discrete system can be efficiently solved by first solving a reduced system for the Lagrange multiplier. Numerical experiments which validate the method are presented.

Given a sequence of finite element spaces which form a de Rham sequence, we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The dual representations also need to satisfy the de Rham sequence on the domain boundary. The matrix which converts primal representations to dual representations – the Hodge matrix – is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even for high order methods. The differential operators for dual representations will be defined. Vector operations, grad, curl and div, for primal and dual representations are both topological and do not depend on the metric, i.e. the size and shape of the mesh or the order of the numerical method. Derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations. As illustration of the use of dual representations, the method will be applied to (i) a mixed formulation for the Poisson problem in 3D, (ii) it will be shown that this approach allows one to preserve the equivalence between Dirichlet and Neumann problems in the finite dimensional setting and, (iii) the method will be applied to the approximation of grad–div eigenvalue problem on affine and non-affine meshes.

The research program of the TCV tokamak ranges from conventional to advanced-tokamak scenarios and alternative divertor configurations, to exploratory plasmas driven by theoretical insight, exploiting the device's unique shaping capabilities. Disruption avoidance by real-time locked mode prevention or unlocking with electron-cyclotron resonance heating (ECRH) was thoroughly documented, using magnetic and radiation triggers. Runaway generation with high-Z noble-gas injection and runaway dissipation by subsequent Ne or Ar injection were studied for model validation. The new 1 MW neutral beam injector has expanded the parameter range, now encompassing ELMy H-modes in an ITER-like shape and nearly non-inductive H-mode discharges sustained by electron cyclotron and neutral beam current drive. In the H-mode, the pedestal pressure increases modestly with nitrogen seeding while fueling moves the density pedestal outwards, but the plasma stored energy is largely uncorrelated to either seeding or fueling. High fueling at high triangularity is key to accessing the attractive small edge-localized mode (type-II) regime. Turbulence is reduced in the core at negative triangularity, consistent with increased confinement and in accord with global gyrokinetic simulations. The geodesic acoustic mode, possibly coupled with avalanche events, has been linked with particle flow to the wall in diverted plasmas. Detachment, scrape-off layer transport, and turbulence were studied in L- and H-modes in both standard and alternative configurations (snowflake, super-X, and beyond). The detachment process is caused by power 'starvation' reducing the ionization source, with volume recombination playing only a minor role. Partial detachment in the H-mode is obtained with impurity seeding and has shown little dependence on flux expansion in standard single-null geometry. In the attached L-mode phase, increasing the outer connection length reduces the in-out heat-flow asymmetry. A doublet plasma, featuring an internal X-point, was achieved successfully, and a transport barrier was observed in the mantle just outside the internal separatrix. In the near future variable-configuration baffles and possibly divertor pumping will be introduced to investigate the effect of divertor closure on exhaust and performance, and 3.5 MW ECRH and 1 MW neutral beam injection heating will be added.

The research program of the TCV tokamak ranges from conventional to advanced-tokamak scenarios and alternative divertor configurations, to exploratory plasmas driven by theoretical insight, exploiting the device's unique shaping capabilities. Disruption avoidance by real-time locked mode prevention or unlocking with electron-cyclotron resonance heating (ECRH) was thoroughly documented, using magnetic and radiation triggers. Runaway generation with high-Z noble-gas injection and runaway dissipation by subsequent Ne or Ar injection were studied for model validation. The new 1 MW neutral beam injector has expanded the parameter range, now encompassing ELMy H-modes in an ITER-like shape and nearly non-inductive H-mode discharges sustained by electron cyclotron and neutral beam current drive. In the H-mode, the pedestal pressure increases modestly with nitrogen seeding while fueling moves the density pedestal outwards, but the plasma stored energy is largely uncorrelated to either seeding or fueling. High fueling at high triangularity is key to accessing the attractive small edge-localized mode (type-II) regime. Turbulence is reduced in the core at negative triangularity, consistent with increased confinement and in accord with global gyrokinetic simulations. The geodesic acoustic mode, possibly coupled with avalanche events, has been linked with particle flow to the wall in diverted plasmas. Detachment, scrape-off layer transport, and turbulence were studied in L- and H-modes in both standard and alternative configurations (snowflake, super-X, and beyond). The detachment process is caused by power 'starvation' reducing the ionization source, with volume recombination playing only a minor role. Partial detachment in the H-mode is obtained with impurity seeding and has shown little dependence on flux expansion in standard single-null geometry. In the attached L-mode phase, increasing the outer connection length reduces the in-out heat-flow asymmetry. A doublet plasma, featuring an internal X-point, was achieved successfully, and a transport barrier was observed in the mantle just outside the internal separatrix. In the near future variable-configuration baffles and possibly divertor pumping will be introduced to investigate the effect of divertor closure on exhaust and performance, and 3.5 MW ECRH and 1 MW neutral beam injection heating will be added.

The research program of the TCV tokamak ranges from conventional to advanced-tokamak scenarios and alternative divertor configurations, to exploratory plasmas driven by theoretical insight, exploiting the device's unique shaping capabilities. Disruption avoidance by real-time locked mode prevention or unlocking with electron-cyclotron resonance heating (ECRH) was thoroughly documented, using magnetic and radiation triggers. Runaway generation with high-Z noble-gas injection and runaway dissipation by subsequent Ne or Ar injection were studied for model validation. The new 1 MW neutral beam injector has expanded the parameter range, now encompassing ELMy H-modes in an ITER-like shape and nearly non-inductive H-mode discharges sustained by electron cyclotron and neutral beam current drive. In the H-mode, the pedestal pressure increases modestly with nitrogen seeding while fueling moves the density pedestal outwards, but the plasma stored energy is largely uncorrelated to either seeding or fueling. High fueling at high triangularity is key to accessing the attractive small edge-localized mode (type-II) regime. Turbulence is reduced in the core at negative triangularity, consistent with increased confinement and in accord with global gyrokinetic simulations. The geodesic acoustic mode, possibly coupled with avalanche events, has been linked with particle flow to the wall in diverted plasmas. Detachment, scrape-off layer transport, and turbulence were studied in L- and H-modes in both standard and alternative configurations (snowflake, super-X, and beyond). The detachment process is caused by power 'starvation' reducing the ionization source, with volume recombination playing only a minor role. Partial detachment in the H-mode is obtained with impurity seeding and has shown little dependence on flux expansion in standard single-null geometry. In the attached L-mode phase, increasing the outer connection length reduces the in-out heat-flow asymmetry. A doublet plasma, featuring an internal X-point, was achieved successfully, and a transport barrier was observed in the mantle just outside the internal separatrix. In the near future variable-configuration baffles and possibly divertor pumping will be introduced to investigate the effect of divertor closure on exhaust and performance, and 3.5 MW ECRH and 1 MW neutral beam injection heating will be added.