Possible solution

#include <cmath>

#include <hpc/matvec/axpy.hpp>
#include <hpc/matvec/copy.hpp>
#include <hpc/matvec/densevector.hpp>
#include <hpc/matvec/iterators.hpp>
#include <hpc/matvec/traits.hpp>
#include <hpc/matvec/print.hpp>

//------------------------------------------------------------------------------

using namespace hpc;
using namespace hpc::matvec;


template <typename T>
const T &
assertEqual(const T &a, const T &b)
{
    assert(a==b);
    return a;
}

//------------------------------------------------------------------------------

template <
    typename F, typename G,
    typename T, template<typename> class VectorF,
                template<typename> class VectorU,
                Require< Dense<VectorF<T>>,
                         Dense<VectorU<T>> > = true
>
void
discretizeProblem(const F &f, const G &g, VectorF<T> &f_h, VectorU<T> &u_h)
{
    auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
    auto h  = T(1)/(N+1);

    u_h(0)   = g(0);
    u_h(N+1) = g(1);

    for (std::size_t i=1; i<N; ++i) {
        u_h(i) = 0;
        f_h(i) = f(i*h);
    }
    // actually we don't have to initialize f_h(0), f_h(N+1) so we store some
    // good and evil numbers there:
    f_h(0)   = 42;
    f_h(N+1) = 666;
}

//------------------------------------------------------------------------------

template <
    typename T, template<typename> class Vector,
                Require< Dense<Vector<T>> > = true
>
T
lInfNorm(const Vector<T> &x_h)
{
    T res = 0;

    for (auto [i, xi] : x_h) {
        i=i;
        if (std::abs(xi) > res) {
            res = std::abs(xi);
        }
    }
    return res;
}

//------------------------------------------------------------------------------

template <
    typename U,
    typename T, template<typename> class VectorU,
                template<typename> class VectorE,
                Require< Dense<VectorU<T>>,
                         Dense<VectorE<T>> > = true
>
void
evalError(const U &u, const VectorU<T> &u_h, VectorE<T> &e_h)
{
    auto N  = assertEqual(u_h.length(), e_h.length()) - 2;
    auto h  = T(1)/(N+1);

    for (auto [i, ei] : e_h) {
        ei = u(i*h) - u_h(i);
    }
}

//------------------------------------------------------------------------------

template <
    typename T, template<typename> class VectorU,
                template<typename> class VectorF,
                template<typename> class VectorR,
                Require< Dense<VectorF<T>>,
                         Dense<VectorU<T>>,
                         Dense<VectorR<T>> > = true
>
void
evalResidual(const VectorU<T> &u_h, const VectorF<T> &f_h, VectorR<T> &r_h)
{
    auto N  = assertEqual(u_h.length(), r_h.length()) - 2;
    auto h  = T(1)/(N+1);
    auto h2 = h*h;

    r_h(0) = r_h(N+1) = 0;
    for (std::size_t i=1; i<=N; ++i) {
        r_h(i) = f_h(i) - 1./h2 * (2*u_h(i) - u_h(i-1) - u_h(i+1));
    }
}


//------------------------------------------------------------------------------

template <
    typename T, template<typename> class VectorF,
                template<typename> class VectorU,
                Require< Dense<VectorF<T>>,
                         Dense<VectorU<T>> > = true
>
void
direct_solver(const VectorF<T> &f_h, VectorU<T> &u_h)
{
    auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
    auto h  = T(1)/(N+1);
    auto h2 = h*h;

    // forward substitution
    u_h(1) = h2*f_h(1) + u_h(0);
    for (std::size_t k=2; k<N; ++k) {
        u_h(k) = h2*f_h(k) + u_h(k-1)*(k-1)/k;
    }
    u_h(N) = h2*f_h(N) + u_h(N+1) + u_h(N-1)*(N-1)/N;

    // backward substitution
    u_h(N) = u_h(N)*N/(N+1);
    for (std::size_t k=N-1; k>=1; --k) {
        u_h(k) = (u_h(k)+u_h(k+1))*k/(k+1);
    }
}

//==============================================================================

template <
    typename T, template<typename> class Vector,
                Require< Dense<Vector<T>> > = true
>
void
addNoise(std::size_t k, Vector<T> &x_h)
{
    auto N  = x_h.length()-2;
    auto h  = T(1)/(N+1);

    for (std::size_t i=1; i<=N; ++i) {
        x_h(i) += sin(M_PI*k*i*h);
    }
}

//------------------------------------------------------------------------------

template <
    typename T, template<typename> class VectorF,
                template<typename> class VectorU,
                template<typename> class VectorS,
                Require< Dense<VectorF<T>>,
                         Dense<VectorU<T>>,
                         Dense<VectorS<T>> > = true
>
void
jacobi_step(double omega, const VectorF<T> &f_h, const VectorU<T> &u_h,
            VectorS<T> &s_h)
{
    auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
    auto h  = T(1)/(N+1);
    auto h2 = h*h;

    for (std::size_t i=1; i<=N; ++i) {
        s_h(i) = (1-omega)*u_h(i) + omega*0.5*(u_h(i-1) +u_h(i+1)+h2*f_h(i));
    }
    s_h(0)   = u_h(0);
    s_h(N+1) = u_h(N+1);
}

//------------------------------------------------------------------------------

int
main()
{
    /*
    auto f = [](double x) { return -2; };
    auto g = [](double x) { return 1 + 3*x; };
    auto u = [](double x) { return (x+1)*(x+1); };
    */
    auto f = [](double x) { return 0; };
    auto g = [](double x) { return 0; };
    auto u = [](double x) { return 0; };

    std::size_t N = 100;

    DenseVector<double>  f_h(N+2), u_h(N+2), tmp_h(N+2), r_h(N+2), e_h(N+2);

    discretizeProblem(f, g, f_h, u_h);

    // First initialize u_h with exact solution
    direct_solver(f_h, u_h);

    // then add some noise 
    //addNoise(1, u_h);
    //addNoise(N/2, u_h);
    //addNoise(N-3, u_h);
    addNoise(N-1, u_h);

    double omega = 0.5;     // smoothing parameter

    for (std::size_t it=0; it<=60; ++it) {
        jacobi_step(omega, f_h, u_h, tmp_h);
        jacobi_step(omega, f_h, tmp_h, u_h);

        evalResidual(u_h, f_h, r_h);
        evalError(u, u_h, e_h);

        fmt::printf("#lInfNorm(r_h) = %.5e\n", lInfNorm(r_h));
        fmt::printf("#lInfNorm(e_h) = %.5e\n", lInfNorm(e_h));
        fmt::printf("it %3d\n", it);
        for (auto [i,ri] : r_h) {
            fmt::printf("%d %f\n", i, ri);
        }
        fmt::printf("\n\n");
    }
}