Function.h

// Copyright (C) 2003-2024 Anders Logg, Garth N. Wells and Massimiliano Leoni
//
// This file is part of DOLFINx (https://www.fenicsproject.org)
//
// SPDX-License-Identifier:    LGPL-3.0-or-later
 
#pragma once
 
#include "DofMap.h"
#include "FiniteElement.h"
#include "FunctionSpace.h"
#include "interpolate.h"
#include <algorithm>
#include <concepts>
#include <dolfinx/common/IndexMap.h>
#include <dolfinx/common/types.h>
#include <dolfinx/la/Vector.h>
#include <dolfinx/mesh/Geometry.h>
#include <dolfinx/mesh/Mesh.h>
#include <dolfinx/mesh/Topology.h>
#include <functional>
#include <memory>
#include <numeric>
#include <span>
#include <string>
#include <utility>
#include <vector>
 
namespace dolfinx::fem
{
template <dolfinx::scalar T, std::floating_point U>
class Expression;
 
template <dolfinx::scalar T,
          std::floating_point U = dolfinx::scalar_value_type_t<T>>
class Function
{
public:
  using value_type = T;
  using geometry_type = U;
 
  explicit Function(std::shared_ptr<const FunctionSpace<geometry_type>> V)
      : _function_space(V),
        _x(std::make_shared<la::Vector<value_type>>(
            V->dofmap()->index_map, V->dofmap()->index_map_bs()))
  {
    if (!V->component().empty())
    {
      throw std::runtime_error("Cannot create Function from subspace. Consider "
                               "collapsing the function space");
    }
  }
 
  Function(std::shared_ptr<const FunctionSpace<geometry_type>> V,
           std::shared_ptr<la::Vector<value_type>> x)
      : _function_space(V), _x(x)
  {
    // NOTE: We do not check for a subspace since this constructor is
    // used for creating subfunctions
 
    // Assertion uses '<=' to deal with sub-functions
    assert(V->dofmap());
    assert(V->dofmap()->index_map->size_global() * V->dofmap()->index_map_bs()
           <= _x->bs() * _x->index_map()->size_global());
  }
 
  // Copy constructor
  Function(const Function& v) = delete;
 
  Function(Function&& v) = default;
 
  ~Function() = default;
 
  Function& operator=(Function&& v) = default;
 
  // Assignment
  Function& operator=(const Function& v) = delete;
 
  Function sub(int i) const
  {
    auto sub_space = std::make_shared<FunctionSpace<geometry_type>>(
        _function_space->sub({i}));
    assert(sub_space);
    return Function(sub_space, _x);
  }
 
  Function collapse() const
  {
    // Create new collapsed FunctionSpace
    auto [V, map] = _function_space->collapse();
 
    // Create new vector
    auto x = std::make_shared<la::Vector<value_type>>(
        V.dofmap()->index_map, V.dofmap()->index_map_bs());
 
    // Copy values into new vector
    std::span<const value_type> x_old = _x->array();
    std::span<value_type> x_new = x->mutable_array();
    for (std::size_t i = 0; i < map.size(); ++i)
    {
      assert((int)i < x_new.size());
      assert(map[i] < x_old.size());
      x_new[i] = x_old[map[i]];
    }
 
    return Function(
        std::make_shared<FunctionSpace<geometry_type>>(std::move(V)), x);
  }
 
  std::shared_ptr<const FunctionSpace<geometry_type>> function_space() const
  {
    return _function_space;
  }
 
  std::shared_ptr<const la::Vector<value_type>> x() const { return _x; }
 
  std::shared_ptr<la::Vector<value_type>> x() { return _x; }
 
  void
  interpolate(const std::function<
              std::pair<std::vector<value_type>, std::vector<std::size_t>>(
                  MDSPAN_IMPL_STANDARD_NAMESPACE::mdspan<
                      const geometry_type,
                      MDSPAN_IMPL_STANDARD_NAMESPACE::extents<
                          std::size_t, 3,
                          MDSPAN_IMPL_STANDARD_NAMESPACE::dynamic_extent>>)>& f)
  {
    assert(_function_space);
    assert(_function_space->mesh());
    int tdim = _function_space->mesh()->topology()->dim();
    auto cmap = _function_space->mesh()->topology()->index_map(tdim);
    assert(cmap);
    std::vector<std::int32_t> cells(cmap->size_local() + cmap->num_ghosts(), 0);
    std::iota(cells.begin(), cells.end(), 0);
    interpolate(f, cells);
  }
 
  void interpolate(
      const std::function<
          std::pair<std::vector<value_type>, std::vector<std::size_t>>(
              MDSPAN_IMPL_STANDARD_NAMESPACE::mdspan<
                  const geometry_type,
                  MDSPAN_IMPL_STANDARD_NAMESPACE::extents<
                      std::size_t, 3,
                      MDSPAN_IMPL_STANDARD_NAMESPACE::dynamic_extent>>)>& f,
      std::span<const std::int32_t> cells)
  {
    assert(_function_space);
    assert(_function_space->element());
    assert(_function_space->mesh());
    const std::vector<geometry_type> x
        = fem::interpolation_coords<geometry_type>(
            *_function_space->element(), _function_space->mesh()->geometry(),
            cells);
    MDSPAN_IMPL_STANDARD_NAMESPACE::mdspan<
        const geometry_type,
        MDSPAN_IMPL_STANDARD_NAMESPACE::extents<
            std::size_t, 3, MDSPAN_IMPL_STANDARD_NAMESPACE::dynamic_extent>>
        _x(x.data(), 3, x.size() / 3);
 
    const auto [fx, fshape] = f(_x);
    assert(fshape.size() <= 2);
    if (int vs = _function_space->element()->value_size();
        vs == 1 and fshape.size() == 1)
    {
      // Check for scalar-valued functions
      if (fshape.front() != x.size() / 3)
        throw std::runtime_error("Data returned by callable has wrong length");
    }
    else
    {
      // Check for vector/tensor value
      if (fshape.size() != 2)
        throw std::runtime_error("Expected 2D array of data");
 
      if (fshape[0] != vs)
      {
        throw std::runtime_error(
            "Data returned by callable has wrong shape(0) size");
      }
 
      if (fshape[1] != x.size() / 3)
      {
        throw std::runtime_error(
            "Data returned by callable has wrong shape(1) size");
      }
    }
 
    std::array<std::size_t, 2> _fshape;
    if (fshape.size() == 1)
      _fshape = {1, fshape[0]};
    else
      _fshape = {fshape[0], fshape[1]};
 
    fem::interpolate(*this, std::span<const value_type>(fx.data(), fx.size()),
                     _fshape, cells);
  }
 
  void interpolate(const Function<value_type, geometry_type>& u)
  {
    assert(_function_space);
    assert(_function_space->mesh());
    int tdim = _function_space->mesh()->topology()->dim();
    auto cmap = _function_space->mesh()->topology()->index_map(tdim);
    assert(cmap);
    std::vector<std::int32_t> cells(cmap->size_local() + cmap->num_ghosts(), 0);
    std::iota(cells.begin(), cells.end(), 0);
    interpolate(u, cells, cells);
  }
 
  void interpolate(const Function<value_type, geometry_type>& u0,
                   std::span<const std::int32_t> cells0,
                   std::span<const std::int32_t> cells1 = {})
  {
    if (cells1.empty())
      cells1 = cells0;
    fem::interpolate(*this, cells1, u0, cells0);
  }
 
  void interpolate(const Expression<value_type, geometry_type>& e)
  {
    assert(_function_space);
    assert(_function_space->mesh());
    int tdim = _function_space->mesh()->topology()->dim();
    auto cmap = _function_space->mesh()->topology()->index_map(tdim);
    assert(cmap);
    std::vector<std::int32_t> cells(cmap->size_local() + cmap->num_ghosts(), 0);
    std::iota(cells.begin(), cells.end(), 0);
    interpolate(e, cells);
  }
 
  void interpolate(const Expression<value_type, geometry_type>& e0,
                   std::span<const std::int32_t> cells0,
                   std::span<const std::int32_t> cells1 = {})
  {
    // Extract mesh
    const mesh::Mesh<geometry_type>* mesh0 = nullptr;
    for (auto& c : e0.coefficients())
    {
      assert(c);
      assert(c->function_space());
      assert(c->function_space()->mesh());
      if (const mesh::Mesh<geometry_type>* mesh
          = c->function_space()->mesh().get();
          !mesh0)
      {
        mesh0 = mesh;
      }
      else if (mesh != mesh0)
      {
        throw std::runtime_error(
            "Expression coefficient Functions have different meshes.");
      }
    }
 
    // If Expression has no Function coefficients take mesh from `this`.
    assert(_function_space);
    assert(_function_space->mesh());
    if (!mesh0)
      mesh0 = _function_space->mesh().get();
 
    // If cells1 is empty and Function and Expression meshes are the
    // same, make cells1 the same as cells0. Otherwise check that
    // lengths of cells0 and cells1 are the same.
    if (cells1.empty() and mesh0 == _function_space->mesh().get())
      cells1 = cells0;
    else if (cells0.size() != cells1.size())
      throw std::runtime_error("Cells lists have different lengths.");
 
    // Check that Function and Expression spaces are compatible
    assert(_function_space->element());
    std::size_t value_size = e0.value_size();
    if (e0.argument_function_space())
      throw std::runtime_error("Cannot interpolate Expression with Argument.");
    if (value_size != _function_space->element()->value_size())
    {
      throw std::runtime_error(
          "Function value size not equal to Expression value size.");
    }
 
    // Compatibility check
    {
      auto [X0, shape0] = e0.X();
      auto [X1, shape1] = _function_space->element()->interpolation_points();
      if (shape0 != shape1)
      {
        throw std::runtime_error(
            "Function element interpolation points has different shape to "
            "Expression interpolation points");
      }
 
      for (std::size_t i = 0; i < X0.size(); ++i)
      {
        if (std::abs(X0[i] - X1[i]) > 1.0e-10)
        {
          throw std::runtime_error("Function element interpolation points not "
                                   "equal to Expression interpolation points");
        }
      }
    }
 
    // Array to hold evaluated Expression
    std::size_t num_cells = cells0.size();
    std::size_t num_points = e0.X().second[0];
    std::vector<value_type> fdata(num_cells * num_points * value_size);
    MDSPAN_IMPL_STANDARD_NAMESPACE::mdspan<
        const value_type,
        MDSPAN_IMPL_STANDARD_NAMESPACE::dextents<std::size_t, 3>>
        f(fdata.data(), num_cells, num_points, value_size);
 
    // Evaluate Expression at points
    e0.eval(*mesh0, cells0, fdata, {num_cells, num_points * value_size});
 
    // Reshape evaluated data to fit interpolate.
    // Expression returns matrix of shape (num_cells, num_points *
    // value_size), i.e. xyzxyz ordering of dof values per cell per
    // point. The interpolation uses xxyyzz input, ordered for all
    // points of each cell, i.e. (value_size, num_cells*num_points).
    std::vector<value_type> fdata1(num_cells * num_points * value_size);
    MDSPAN_IMPL_STANDARD_NAMESPACE::mdspan<
        value_type, MDSPAN_IMPL_STANDARD_NAMESPACE::dextents<std::size_t, 3>>
        f1(fdata1.data(), value_size, num_cells, num_points);
    for (std::size_t i = 0; i < f.extent(0); ++i)
      for (std::size_t j = 0; j < f.extent(1); ++j)
        for (std::size_t k = 0; k < f.extent(2); ++k)
          f1(k, i, j) = f(i, j, k);
 
    // Interpolate values into appropriate space
    fem::interpolate(*this,
                     std::span<const value_type>(fdata1.data(), fdata1.size()),
                     {value_size, num_cells * num_points}, cells1);
  }
 
  void interpolate(const Function<value_type, geometry_type>& v,
                   std::span<const std::int32_t> cells,
                   const geometry::PointOwnershipData<U>& interpolation_data)
  {
    fem::interpolate(*this, v, cells, interpolation_data);
  }
 
  void eval(std::span<const geometry_type> x, std::array<std::size_t, 2> xshape,
            std::span<const std::int32_t> cells, std::span<value_type> u,
            std::array<std::size_t, 2> ushape) const
  {
    if (cells.empty())
      return;
 
    assert(x.size() == xshape[0] * xshape[1]);
    assert(u.size() == ushape[0] * ushape[1]);
 
    // TODO: This could be easily made more efficient by exploiting
    // points being ordered by the cell to which they belong.
 
    if (xshape[0] != cells.size())
    {
      throw std::runtime_error(
          "Number of points and number of cells must be equal.");
    }
 
    if (xshape[0] != ushape[0])
    {
      throw std::runtime_error(
          "Length of array for Function values must be the "
          "same as the number of points.");
    }
 
    // Get mesh
    assert(_function_space);
    auto mesh = _function_space->mesh();
    assert(mesh);
    const std::size_t gdim = mesh->geometry().dim();
    const std::size_t tdim = mesh->topology()->dim();
    auto map = mesh->topology()->index_map(tdim);
 
    // Get coordinate map
    const CoordinateElement<geometry_type>& cmap = mesh->geometry().cmap();
 
    // Get geometry data
    auto x_dofmap = mesh->geometry().dofmap();
    const std::size_t num_dofs_g = cmap.dim();
    auto x_g = mesh->geometry().x();
 
    // Get element
    auto element = _function_space->element();
    assert(element);
    const int bs_element = element->block_size();
    const std::size_t reference_value_size = element->reference_value_size();
    const std::size_t value_size
        = _function_space->element()->reference_value_size();
    const std::size_t space_dimension = element->space_dimension() / bs_element;
 
    // If the space has sub elements, concatenate the evaluations on the
    // sub elements
    const int num_sub_elements = element->num_sub_elements();
    if (num_sub_elements > 1 and num_sub_elements != bs_element)
    {
      throw std::runtime_error("Function::eval is not supported for mixed "
                               "elements. Extract subspaces.");
    }
 
    // Create work vector for expansion coefficients
    std::vector<value_type> coefficients(space_dimension * bs_element);
 
    // Get dofmap
    std::shared_ptr<const DofMap> dofmap = _function_space->dofmap();
    assert(dofmap);
    const int bs_dof = dofmap->bs();
 
    std::span<const std::uint32_t> cell_info;
    if (element->needs_dof_transformations())
    {
      mesh->topology_mutable()->create_entity_permutations();
      cell_info = std::span(mesh->topology()->get_cell_permutation_info());
    }
 
    std::vector<geometry_type> coord_dofs_b(num_dofs_g * gdim);
    impl::mdspan_t<geometry_type, 2> coord_dofs(coord_dofs_b.data(), num_dofs_g,
                                                gdim);
    std::vector<geometry_type> xp_b(1 * gdim);
    impl::mdspan_t<geometry_type, 2> xp(xp_b.data(), 1, gdim);
 
    // Loop over points
    std::ranges::fill(u, 0.0);
    std::span<const value_type> _v = _x->array();
 
    // Evaluate geometry basis at point (0, 0, 0) on the reference cell.
    // Used in affine case.
    std::array<std::size_t, 4> phi0_shape = cmap.tabulate_shape(1, 1);
    std::vector<geometry_type> phi0_b(std::reduce(
        phi0_shape.begin(), phi0_shape.end(), 1, std::multiplies{}));
    impl::mdspan_t<const geometry_type, 4> phi0(phi0_b.data(), phi0_shape);
    cmap.tabulate(1, std::vector<geometry_type>(tdim), {1, tdim}, phi0_b);
    auto dphi0 = MDSPAN_IMPL_STANDARD_NAMESPACE::submdspan(
        phi0, std::pair(1, tdim + 1), 0,
        MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent, 0);
 
    // Data structure for evaluating geometry basis at specific points.
    // Used in non-affine case.
    std::array<std::size_t, 4> phi_shape = cmap.tabulate_shape(1, 1);
    std::vector<geometry_type> phi_b(
        std::reduce(phi_shape.begin(), phi_shape.end(), 1, std::multiplies{}));
    impl::mdspan_t<const geometry_type, 4> phi(phi_b.data(), phi_shape);
    auto dphi = MDSPAN_IMPL_STANDARD_NAMESPACE::submdspan(
        phi, std::pair(1, tdim + 1), 0,
        MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent, 0);
 
    // Reference coordinates for each point
    std::vector<geometry_type> Xb(xshape[0] * tdim);
    impl::mdspan_t<geometry_type, 2> X(Xb.data(), xshape[0], tdim);
 
    // Geometry data at each point
    std::vector<geometry_type> J_b(xshape[0] * gdim * tdim);
    impl::mdspan_t<geometry_type, 3> J(J_b.data(), xshape[0], gdim, tdim);
    std::vector<geometry_type> K_b(xshape[0] * tdim * gdim);
    impl::mdspan_t<geometry_type, 3> K(K_b.data(), xshape[0], tdim, gdim);
    std::vector<geometry_type> detJ(xshape[0]);
    std::vector<geometry_type> det_scratch(2 * gdim * tdim);
 
    // Prepare geometry data in each cell
    for (std::size_t p = 0; p < cells.size(); ++p)
    {
      const int cell_index = cells[p];
 
      // Skip negative cell indices
      if (cell_index < 0)
        continue;
 
      // Get cell geometry (coordinate dofs)
      auto x_dofs = MDSPAN_IMPL_STANDARD_NAMESPACE::submdspan(
          x_dofmap, cell_index, MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent);
      assert(x_dofs.size() == num_dofs_g);
      for (std::size_t i = 0; i < num_dofs_g; ++i)
      {
        const int pos = 3 * x_dofs[i];
        for (std::size_t j = 0; j < gdim; ++j)
          coord_dofs(i, j) = x_g[pos + j];
      }
 
      for (std::size_t j = 0; j < gdim; ++j)
        xp(0, j) = x[p * xshape[1] + j];
 
      auto _J = MDSPAN_IMPL_STANDARD_NAMESPACE::submdspan(
          J, p, MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent,
          MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent);
      auto _K = MDSPAN_IMPL_STANDARD_NAMESPACE::submdspan(
          K, p, MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent,
          MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent);
 
      std::array<geometry_type, 3> Xpb = {0, 0, 0};
      MDSPAN_IMPL_STANDARD_NAMESPACE::mdspan<
          geometry_type,
          MDSPAN_IMPL_STANDARD_NAMESPACE::extents<
              std::size_t, 1, MDSPAN_IMPL_STANDARD_NAMESPACE::dynamic_extent>>
          Xp(Xpb.data(), 1, tdim);
 
      // Compute reference coordinates X, and J, detJ and K
      if (cmap.is_affine())
      {
        CoordinateElement<geometry_type>::compute_jacobian(dphi0, coord_dofs,
                                                           _J);
        CoordinateElement<geometry_type>::compute_jacobian_inverse(_J, _K);
        std::array<geometry_type, 3> x0 = {0, 0, 0};
        for (std::size_t i = 0; i < coord_dofs.extent(1); ++i)
          x0[i] += coord_dofs(0, i);
        CoordinateElement<geometry_type>::pull_back_affine(Xp, _K, x0, xp);
        detJ[p]
            = CoordinateElement<geometry_type>::compute_jacobian_determinant(
                _J, det_scratch);
      }
      else
      {
        // Pull-back physical point xp to reference coordinate Xp
        cmap.pull_back_nonaffine(Xp, xp, coord_dofs);
        cmap.tabulate(1, std::span(Xpb.data(), tdim), {1, tdim}, phi_b);
        CoordinateElement<geometry_type>::compute_jacobian(dphi, coord_dofs,
                                                           _J);
        CoordinateElement<geometry_type>::compute_jacobian_inverse(_J, _K);
        detJ[p]
            = CoordinateElement<geometry_type>::compute_jacobian_determinant(
                _J, det_scratch);
      }
 
      for (std::size_t j = 0; j < X.extent(1); ++j)
        X(p, j) = Xpb[j];
    }
 
    // Prepare basis function data structures
    std::vector<geometry_type> basis_derivatives_reference_values_b(
        1 * xshape[0] * space_dimension * reference_value_size);
    impl::mdspan_t<const geometry_type, 4> basis_derivatives_reference_values(
        basis_derivatives_reference_values_b.data(), 1, xshape[0],
        space_dimension, reference_value_size);
    std::vector<geometry_type> basis_values_b(space_dimension * value_size);
    impl::mdspan_t<geometry_type, 2> basis_values(basis_values_b.data(),
                                                  space_dimension, value_size);
 
    // Compute basis on reference element
    element->tabulate(basis_derivatives_reference_values_b, Xb,
                      {X.extent(0), X.extent(1)}, 0);
 
    using xu_t = impl::mdspan_t<geometry_type, 2>;
    using xU_t = impl::mdspan_t<const geometry_type, 2>;
    using xJ_t = impl::mdspan_t<const geometry_type, 2>;
    using xK_t = impl::mdspan_t<const geometry_type, 2>;
    auto push_forward_fn
        = element->basix_element().template map_fn<xu_t, xU_t, xJ_t, xK_t>();
 
    // Transformation function for basis function values
    auto apply_dof_transformation
        = element->template dof_transformation_fn<geometry_type>(
            doftransform::standard);
 
    // Size of tensor for symmetric elements, unused in non-symmetric case, but
    // placed outside the loop for pre-computation.
    int matrix_size;
    if (element->symmetric())
    {
      matrix_size = 0;
      while (matrix_size * matrix_size < ushape[1])
        ++matrix_size;
    }
 
    const std::size_t num_basis_values = space_dimension * reference_value_size;
    for (std::size_t p = 0; p < cells.size(); ++p)
    {
      const int cell_index = cells[p];
      if (cell_index < 0) // Skip negative cell indices
        continue;
 
      // Permute the reference basis function values to account for the
      // cell's orientation
      apply_dof_transformation(
          std::span(basis_derivatives_reference_values_b.data()
                        + p * num_basis_values,
                    num_basis_values),
          cell_info, cell_index, reference_value_size);
 
      {
        auto _U = MDSPAN_IMPL_STANDARD_NAMESPACE::submdspan(
            basis_derivatives_reference_values, 0, p,
            MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent,
            MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent);
        auto _J = MDSPAN_IMPL_STANDARD_NAMESPACE::submdspan(
            J, p, MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent,
            MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent);
        auto _K = MDSPAN_IMPL_STANDARD_NAMESPACE::submdspan(
            K, p, MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent,
            MDSPAN_IMPL_STANDARD_NAMESPACE::full_extent);
        push_forward_fn(basis_values, _U, _J, detJ[p], _K);
      }
 
      // Get degrees of freedom for current cell
      std::span<const std::int32_t> dofs = dofmap->cell_dofs(cell_index);
      for (std::size_t i = 0; i < dofs.size(); ++i)
        for (int k = 0; k < bs_dof; ++k)
          coefficients[bs_dof * i + k] = _v[bs_dof * dofs[i] + k];
 
      if (element->symmetric())
      {
        int row = 0;
        int rowstart = 0;
        // Compute expansion
        for (int k = 0; k < bs_element; ++k)
        {
          if (k - rowstart > row)
          {
            row++;
            rowstart = k;
          }
          for (std::size_t i = 0; i < space_dimension; ++i)
          {
            for (std::size_t j = 0; j < value_size; ++j)
            {
              u[p * ushape[1]
                + (j * bs_element + row * matrix_size + k - rowstart)]
                  += coefficients[bs_element * i + k] * basis_values(i, j);
              if (k - rowstart != row)
              {
                u[p * ushape[1]
                  + (j * bs_element + row + matrix_size * (k - rowstart))]
                    += coefficients[bs_element * i + k] * basis_values(i, j);
              }
            }
          }
        }
      }
      else
      {
        // Compute expansion
        for (int k = 0; k < bs_element; ++k)
        {
          for (std::size_t i = 0; i < space_dimension; ++i)
          {
            for (std::size_t j = 0; j < value_size; ++j)
            {
              u[p * ushape[1] + (j * bs_element + k)]
                  += coefficients[bs_element * i + k] * basis_values(i, j);
            }
          }
        }
      }
    }
  }
 
  std::string name = "u";
 
private:
  // Function space
  std::shared_ptr<const FunctionSpace<geometry_type>> _function_space;
 
  // Vector of expansion coefficients (local)
  std::shared_ptr<la::Vector<value_type>> _x;
};
 
} // namespace dolfinx::fem